A function  $f$  is defined on the set of all real numbers except  $0$  and takes all real values except  $1$ . It is also known that $\color{white}\ . \ \color{black}\ \quad f(xy)=f(x)f(-y)-f(x)+f(y)$ for any  $x,y\not= 0$  and that $\color{white}\ . \ \color{black}\ \quad f(f(x))=\frac{1}{f(\frac{1}{x})}$ for any  $x\not\in\{ 0,1\}$ . Determine all such functions  $f$ .

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


For any positive integer  $n$  , let  $s(n)$  denote the number of ordered pairs  $(x,y)$  of positive integers for which  $\dfrac{1}{x} + \dfrac{1}{y} = \dfrac{1}{n}$  . Determine the set of positive integers for which  $s(n) = 5$

Solution: \boxed{\{p^2 \mid p \text{ is a prime number}\}}


A point-like mass moves horizontally between two walls on a frictionless surface with initial kinetic energy  E . With every collision with the walls, the mass loses 1/2 of its kinetic energy to thermal energy. Determine the number of collisions necessary before the speed of the mass is reduced by a factor of 8 .


Let  $f(x)=\tfrac{x+a}{x+b}$  for real numbers  $x$  such that  $x\neq -b$ . Compute all pairs of real numbers  $(a,b)$  such that  $f(f(x))=-\tfrac{1}{x}$  for  $x\neq0$ .

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


Let  $ p_1, p_2, p_3$  and  $ p_4$  be four different prime numbers satisying the equations

                    $ 2p_1 \plus{} 3p_2 \plus{} 5p_3 \plus{} 7p_4 \equal{} 162$
                  $ 11p_1 \plus{} 7p_2 \plus{} 5p_3 \plus{} 4p_4 \equal{} 162$

Find all possible values of the product  $ p_1p_2p_3p_4$


Determine all positive integers  $n$  for which there exist positive integers  $a_1,a_2, ..., a_n$

with  $a_1 + 2a_2 + 3a_3 +... + na_n = 6n$  and  $\frac{1}{a_1}+\frac{2}{a_2}+\frac{3}{a_3}+ ... +\frac{n}{a_n}= 2 + \frac1n$

Solution: \boxed{n = 3}


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


Let  $ ABCD$  be a unit square (that is, the labels  $ A, B, C, D$  appear in that order around the square). Let  $ X$  be a point outside of the square such that the distance from  $ X$  to  $ AC$  is equal to the distance from  $ X$  to  $ BD$ , and also that  $ AX \equal{} \frac {\sqrt {2}}{2}$ .  Determine the value of  $ CX^2$ .


$n$  balls are placed independently uniformly at random into  $n$  boxes. One box is selected at random, and is found to contain  $b$  balls. Let  $e_n$  be the expected value of  $b^4$ . Find   $$ \lim_{n \to
\infty}e_n. $$


Let  $f$  be a real-valued function defined over ordered pairs of integers such that \[f(x+3m-2n, y-4m+5n) = f(x,y)\] for every integers  $x,y,m,n$. Determine the maximum possible number of distinct elements in the range set of  $f$.


nic $\kappa$ y is drawing kappas in the cells of a square grid. However, he does not want to draw kappas in three consecutive cells (horizontally, vertically, or diagonally). Find all real numbers  $d>0$  such that for every positive integer  $n,$  nic $\kappa$ y can label at least  $dn^2$  cells of an  $n\times n$  square.

*Proposed by Mihir Singhal and Michael Kural*

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


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

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

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


Inside the  $7\times8$  rectangle below, one point is chosen a distance  $\sqrt2$  from the left side and a distance  $\sqrt7$  from the bottom side. The line segments from that point to the four vertices of the rectangle are drawn. Find the area of the shaded region.
[asy]
import graph;
size(4cm);
pair A = (0,0);
pair B = (9,0);
pair C = (9,7);
pair D = (0,7);
pair P = (1.5,3);
draw(A--B--C--D--cycle,linewidth(1.5));
filldraw(A--B--P--cycle,rgb(.76,.76,.76),linewidth(1.5));
filldraw(C--D--P--cycle,rgb(.76,.76,.76),linewidth(1.5));
[/asy]


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


Compute the minimum value of  $cos(a-b) + cos(b-c) + cos(c-a)$  as  $a,b,c$   ranges over the real numbers.


Charlie plans to sell bananas for forty cents and apples for fifty cents at his fruit stand, but Dave accidentally reverses the prices. After selling all their fruit they earn a dollar more than they would have with the original prices. Find the difference in the number of bananas and apples sold.


**i.)** Let  $g(x) = x^5 + x^4 + x^3 + x^2 + x + 1.$  What is the remainder when the polynomial  $g(x^{12}$  is divided by the polynomial  $g(x)$ ?

**ii.)** If  $k$  is a positive number and  $f$  is a function such that, for every positive number   $x, f(x^2 + 1 )^{\sqrt{x}} = k.$  Find the value of

\[ f( \frac{9 +y^2}{y^2})^{\sqrt{ \frac{12}{y} }} \] for every positive number  $y.$

**iii.)** The function  $f$  satisfies the functional equation  $f(x) + f(y) = f(x+y) - x \cdot y - 1$  for every pair  $x,y$  of real numbers. If  $f(1) = 1,$  then find the numbers of integers  $n,$  for which  $f(n) = n.$


Given a regular octahedron formed by joining the centers of adjoining faces of a cube, determine the ratio of the volume of the octahedron to the volume of the cube.


The expression \((x \plus{} y \plus{} z)^{2006} \plus{} (x \minus{} y \minus{} z)^{2006}\) is simplified by expanding it and combining like terms. How many terms are in the simplified expression?


Determine all pairs  $ (a,b)$  of real numbers such that  $ 10, a, b, ab$  is an arithmetic progression.

Solution: \boxed{(4, -2) \text{ or } (2.5, -5)}


Given that $(x^y)^z=64$, determine the number of ordered triples of positive integers $(x,y,z)$.


The sequence  $\left(a_n \right)$  is defined by  $a_1=1, \ a_2=2$  and  $$ a_{n+2} = 2a_{n+1}-pa_n, \ \forall n \ge 1, $$   for some prime  $p.$  Find all  $p$  for which there exists  $m$  such that  $a_m=-3.$

Solution: \boxed{p = 7}


Let  $ABCD$  be a parallelogram such that  $\angle BAD = 60^{\circ}.$  Let  $K$  and  $L$  be the midpoints of  $BC$  and  $CD,$  respectively. Assuming that  $ABKL$  is a cyclic quadrilateral, find  $\angle ABD.$


Answer the following questions:

(1) Let  $a$  be positive real number. Find  $\lim_{n\to\infty} (1+a^{n})^{\frac{1}{n}}.$ (2) Evaluate  $\int_1^{\sqrt{3}} \frac{1}{x^2}\ln \sqrt{1+x^2}dx.$ 35 points

Solution: \boxed{\left( \frac{1}{2} - \frac{1}{\sqrt{3}} \right) \ln 2 + \frac{\pi}{12}}


Given that the ratios $\dfrac{x}{n}$, $\dfrac{y}{n+1}$, and $\dfrac{z}{n+2}$ are equal, where $x+y+z=90$, determine the number of triples of positive integers $(x,y,z)$.


Marisa has a collection of  $2^8-1=255$  distinct nonempty subsets of  $\{1, 2, 3, 4, 5, 6, 7, 8\}$ . For each step she takes two subsets chosen uniformly at random from the collection, and replaces them with either their union or their intersection, chosen randomly with equal probability. (The collection is allowed to contain repeated sets.) She repeats this process  $2^8-2=254$  times until there is only one set left in the collection. What is the expected size of this set?


Find all triples of natural  $ (x,y,n)$  satisfying the condition:
\[ \frac {x! \plus{} y!}{n!} \equal{} 3^n
\]

Define  $ 0! \equal{} 1$

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


Find the number of integers  $n$  with  $1\le n\le 100$  for which  $n-\phi(n)$  is prime.  Here  $\phi(n)$  denotes the number of positive integers less than  $n$  which are relatively prime to  $n$ .

*Proposed by Mehtaab Sawhney*


Given that Yan rides seven times as fast as he walks and that walking directly to the stadium and walking home then riding a bicycle to the stadium require the same amount of time. If the distance from Yan's current position to his home is represented by $d_h$ and the distance from Yan's current position to the stadium is represented by $d_s$, find the ratio of $d_h$ to $d_s$.


A round table has room for n diners (  $n\ge  2$ ). There are napkins in three different colours. In how many ways can the napkins be placed, one for each seat, so that no two neighbours get napkins of the same colour?

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


Consider the sequence of six real numbers 60, 10, 100, 150, 30, and  $x$ .  The average (arithmetic mean) of this sequence is equal to the median of the sequence.  What is the sum of all the possible values of  $x$ ?  (The median of a sequence of six real numbers is the average of the two middle numbers after all the numbers have been arranged in increasing order.)


BdMO National 2016 Higher Secondary

<u>**Problem 4:**</u>
Consider the set of integers  $  \left \{ 1, 2, ......... , 100 \right \} $ . Let  $   \left \{ x_1, x_2, ......... , x_{100} \right \}$  be some arbitrary arrangement of the integers  $ \left \{ 1, 2, ......... , 100 \right \}$ , where all of the  $x_i$  are different. Find the smallest possible value of the sum, $S = \left | x_2 - x_1 \right | + \left | x_3 - x_2  \right | + ................+ \left |x_{100} - x_{99} \right | + \left |x_1 - x_{100} \right | $ .


$ABCD$  is a square and  $AB = 1$ . Equilateral triangles  $AYB$  and  $CXD$  are drawn such that  $X$  and  $Y$  are inside the square. What is the length of  $XY$ ?

Solution: \boxed{\sqrt{3} - 1}


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

Solution: \boxed{m = 4}


Carl, James, Saif, and Ted play several games of two-player For The Win on the Art of Problem Solving website. If, among these games, Carl wins  $5$  and loses  $0,$  James wins  $4$  and loses  $2,$  Saif wins  $1$  and loses  $6,$  and Ted wins  $4,$  how many games does Ted lose?


The three medians of a triangle has lengths  $3, 4, 5$ . What is the length of the shortest side of this triangle?


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


Let  $R$  be the set of points  $(x, y)$  such that  $\lfloor x^2 \rfloor = \lfloor y \rfloor$  and  $\lfloor y^2 \rfloor = \lfloor x \rfloor$ .  Compute the area of region  $R$ .  Recall that  $\lfloor z \rfloor$  is the greatest integer that is less than or equal to  $z$ .

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


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


What is the maximal number of regions a circle can be divided in by segments joining  $n$  points on the boundary of the circle ?

*Posted already on the board I think...*

Solution: \boxed{\binom{n}{4} + \binom{n}{2} + 1}


Compute the smallest positive integer that is  $3$  more than a multiple of  $5$ , and twice a multiple of  $6$ .


Consider non-negative real numbers  $a, b, c$  satisfying the condition  $a^2 + b^2 + c^2 = 2$  . Find the maximum value of the following expression   $$ P=\frac{\sqrt{b^2+c^2}}{3-a}+\frac{\sqrt{c^2+a^2}}{3-b}+a+b-2022c $$


Find the sum of all integers  $x$ ,  $x \ge 3$ , such that  $201020112012_x$  (that is,  $201020112012$  interpreted as a base  $x$  number) is divisible by  $x-1$


Find the smallest positive integer, such that the sum and the product of it and  $10$  are both square numbers.


Find the sum of the roots of  $\tan^2x-9\tan x+1=0$  that are between  $x=0$  and  $x=2\pi$  radians.


Deﬁne the determinant  $D_1$  =  $|1|$ , the determinant  $D_2$  =      $|1 1|$   $|1 3|$
, and the determinant  $D_3=$
|1 1 1|
|1 3 3|
|1 3 5|
.
In general, for positive integer n, let the determinant  $D_n$  have 1s in every position of its ﬁrst row and ﬁrst
column, 3s in the remaining positions of the second row and second column, 5s in the remaining positions of the third row and third column, and so forth. Find the least n so that  $D_n$   $\geq$  2015.


Find the largest constant  $c>0$  such that for every positive integer  $n\ge 2$ , there always exist a positive divisor  $d$  of  $n$  such that   $$ d\le \sqrt{n}\hspace{0.5cm} \text{and} \hspace{0.5cm} \tau(d)\ge c\sqrt{\tau(n)} $$   where  $\tau(n)$  is the number of divisors of  $n$ .

*Proposed by Mohd. Suhaimi Ramly*


Let  $f$  be the quadratic function with leading coefficient  $1$  whose graph is tangent to that of the lines  $y=-5x+6$  and  $y=x-1$ .  The sum of the coefficients of  $f$  is  $\tfrac pq$ , where  $p$  and  $q$  are positive relatively prime integers.  Find  $100p + q$ .

*Proposed by David Altizio*


Let  $T = \{9^k : k \ \text{is an integer}, 0 \le k \le 4000\}$ .  Given that  $9^{4000}$  has 3817 digits and that its first (leftmost) digit is 9, how many elements of  $T$  have 9 as their leftmost digit?


Given that one must cover a  $k\times k$  square with  $T$  tetrominos, find the sum of all  $k\leq25$  such that  $k^{2}=T$.


Given a continuous function  $ f(x)$  such that  $ \int_0^{2\pi} f(x)\ dx \equal{} 0$ .

Let  $ S(x) \equal{} A_0 \plus{} A_1\cos x \plus{} B_1\sin x$ , find constant numbers  $ A_0,\ A_1$  and  $ B_1$  for which  $ \int_0^{2\pi} \{f(x) \minus{} S(x)\}^2\ dx$  is minimized.

Solution: \boxed{A_0 = 0, A_1 = \frac{1}{\pi} \int_0^{2\pi} f(x) \cos x \, dx, B_1 = \frac{1}{\pi} \int_0^{2\pi} f(x) \sin x \, dx}


Find the sum of all the digits in the decimal representations of all the positive integers less than  $1000.$


For positive real nubers  $a,b,c$  find the maximum real number  $x$ , such that there exist positive numbers  $p,q,r$ , such that  $p+q+r=1$  and  $x$  does not exceed numbers  $a\frac{p}{q}, b\frac{q}{r}, c\frac{r}{p}$


Let  $AB$  be diameter of a circle  $\omega$  and let  $C$  be a point on  $\omega$ , different from  $A$  and  $B$ . The perpendicular from  $C$  intersects  $AB$  at  $D$  and  $\omega$  at  $E(\neq C)$ . The circle with centre at  $C$  and radius  $CD$  intersects  $\omega$  at  $P$  and  $Q$ . If the perimeter of the triangle  $PEQ$  is  $24$ , find the length of the side  $PQ$


Let  $PROBLEMZ$  be a regular octagon inscribed in a circle of unit radius. Diagonals  $MR$ ,  $OZ$  meet at  $I$ . Compute  $LI$ .


There are positive integers  $b$  and  $c$  such that the polynomial  $2x^2 + bx + c$  has two real roots which differ by  $30.$  Find the least possible value of  $b + c.$


Let  $ABC$  be a triangle with centroid  $G$ . Determine, with proof, the position of the point  $P$  in the plane of  $ABC$  such that  $AP{\cdot}AG + BP{\cdot}BG + CP{\cdot}CG$  is a minimum, and express this minimum value in terms of the side lengths of  $ABC$ .

Solution: \boxed{\frac{a^2 + b^2 + c^2}{3}}


Line segment  $\overline{AB}$  has perpendicular bisector  $\overline{CD}$ , where  $C$  is the midpoint of  $\overline{AB}$ . The segments have lengths  $AB = 72$  and  $CD = 60$ . Let  $R$  be the set of points  $P$  that are midpoints of line segments  $\overline{XY}$  , where  $X$  lies on  $\overline{AB}$  and  $Y$  lies on  $\overline{CD}$ . Find the area of the region  $R$ .


Four unit circles are placed on a square of side length  $2$ , with each circle centered on one of the four corners of the square. Compute the area of the square which is not contained within any of the four circles.

Solution: \boxed{4 - \pi}


A particle starts at  $(0,0,0)$  in three-dimensional space. Each second, it randomly selects one of the eight lattice points a distance of  $\sqrt{3}$  from its current location and moves to that point. What is the probability that, after two seconds, the particle is a distance of  $2\sqrt{2}$  from its original location?

*Proposed by Connor Gordon*


For integer  $n$ , let  $I_n=\int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \frac{\cos (2n+1)x}{\sin x}\  dx.$ (1) Find  $I_0.$ (2) For each positive integer  $n$ , find  $I_n-I_{n-1}.$ (3) Find  $I_5$ .

Solution: \boxed{\frac{1}{2} \ln 2 - \frac{8}{15}}


Let  $n$  be square with 4 digits, such that all its digits are less than 6. If we add 1 to each digit the resulting number is another square. Find  $n$


Find all triples  $(a,b,c)$  of real numbers all different from zero that satisfies:

\begin{eqnarray}  a^4+b^2c^2=16a\nonumber  b^4+c^2a^2=16b \nonumber c^4+a^2b^2=16c \nonumber \end{eqnarray}

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


Find the point in the closed unit disc  $D=\{ (x,y) | x^2+y^2\le 1 \}$  at which the function  $f(x,y)=x+y$  attains its maximum .

Solution: \boxed{\left( \frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}} \right)}


Let  $\omega$  be a circle with radius  $1$ . Equilateral triangle  $\vartriangle ABC$  is tangent to  $\omega$   at the midpoint of side  $BC$  and  $\omega$   lies outside   $\vartriangle ABC$ . If line  $AB$  is tangent to  $\omega$  , compute the side length of   $\vartriangle  ABC$ .

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


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  $y.$


Given that a positive number is singular if its representation as a product of powers of distinct prime numbers contains no even powers other than $0$, determine the maximum number of consecutive singular numbers.


Given a polynomial  $P$ with integer coefficients that satisfies  $P\left(0\right)=P\left(2\right)=P\left(5\right)=P\left(6\right)=30$ , determine the largest positive integer  $d$  that is a divisor of  $P\left(n\right)$  for all integers  $n$ .


Suppose  $a, b$  are integers and  $a+b$  is a root of  $x^2 +ax+b = 0$ . What is the maximum possible value of  $b^2$ ?


Find all integers  $k$  such that both  $k + 1$  and  $16k + 1$  are perfect squares.

Solution: \boxed{k = 0 \text{ and } k = 3}


Nine lines parallel to the base of a triangle divide the other sides each into  $10$  equal segments and the area into  $10$  distinct parts. If the area of the largest of these parts is  $38$, calculate the area of the original triangle.


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

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


Determine all polynomials  $P(x) \in \mathbb R[x]$  satisfying the following two conditions :
(a)  $P(2017) = 2016$  and
(b)  $(P(x) + 1)^2 = P(x^2 + 1)$  for all real numbers  $x$ .

*proposed by Walther Janous*

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


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


In tetrahedron  $ABCD$ , edge  $AB$  has length 3 cm.  The area of face  $ABC$  is 15  $\text{cm}^2$  and the area of face  $ABD$  is 12  $\text{cm}^2$ .  These two faces meet each other at a  $30^\circ$  angle.  Find the volume of the tetrahedron in  $\text{cm}^3$ .


Calculate the following indefinite integrals.

[1]  $\int \sin x\sin 2x dx$

[2]  $\int \frac{e^{2x}}{e^x-1}dx$

[3]  $\int \frac{\tan ^2 x}{\cos ^2 x}dx$

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

[5]  $\int \frac{e^x}{e^x+1}dx$

Solution: \boxed{\ln |e^x + 1| + C}


Given a trapezoid  $ABCD$  with bases  $BC$  and  $AD$ , with  $AD=2 BC$ . Let  $M$  be the midpoint of  $AD, E$  be the intersection point of the sides  $AB$  and  $CD$ ,   $O$  be the intersection point of   $BM$  and  $AC, N$  be the intersection point of   $EO$  and  $BC$ . In what ratio, point  $N$  divides the segment  $BC$ ?

Solution: \boxed{BN:NC = 1:2}


For natural numbers  $x$  and  $y$ , let  $(x,y)$  denote the greatest common divisor of  $x$  and  $y$ . How many pairs of natural numbers  $x$  and  $y$  with  $x \le y$  satisfy the equation  $xy = x + y + (x, y)$ ?


The system of equations
$a+bc=1,$
$b+ac=1,$
$c+ab=1,$
has how many real solutions for $a$, $b$, and $c$.


Find all triples of positive integers  $(x, y, z)$  with   $$ \frac{xy}{z}+ \frac{yz}{x}+\frac{zx}{y}= 3 $$

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


In rectangle  $ ABCD$ ,  $ AB\equal{}100$ . Let  $ E$  be the midpoint of  $ \overline{AD}$ . Given that line  $ AC$  and line  $ BE$  are perpendicular, find the greatest integer less than  $ AD$ .


Consider a triangle  $ABC$  with  $BC = 3$ . Choose a point  $D$  on  $BC$  such that  $BD = 2$ . Find the value of
\[AB^2 + 2AC^2 - 3AD^2.\]


A finite set  $\mathcal{S}$  of distinct real numbers has the following properties: the mean of  $\mathcal{S}\cup\{1\}$  is 13 less than the mean of  $\mathcal{S}$ , and the mean of  $\mathcal{S}\cup\{2001\}$  is 27 more than the mean of  $\mathcal{S}.$   Find the mean of  $\mathcal{S}.$


Let  $ \{a_n\}_{n=1}^\infty $  be an arithmetic progression  with  $ a_1 > 0 $  and  $ 5\cdot a_{13} = 6\cdot a_{19} $  . What is the smallest integer  $ n$  such that  $ a_n<0 $ ?


Find the maximal positive integer  $n$ , so that for any real number  $x$  we have  $\sin^{n}{x}+\cos^{n}{x} \geq \frac{1}{n}$ .

Solution: \boxed{n = 8}


Tamika selects two different numbers at random from the set  $ \{ 8,9,10\} $  and adds them. Carlos takes two different numbers at random from the set  $ \{ 3,5,6\} $  and multiplies them. Calculate the probability that Tamika's result is greater than Carlos' result.


Points  $A_1, A_2, \ldots, A_{2022}$  are chosen on a plane so that no three of them are collinear. Consider all angles  $A_iA_jA_k$  for distinct points  $A_i, A_j, A_k$ . What largest possible number of these angles can be equal to  $90^\circ$ ?

*Proposed by Anton Trygub*


There is the sequence of numbers  $1, a_2, a_3, ...$  such that satisfies  $1 \cdot a_2  \cdot a_3 \cdot  ...  \cdot a_n = n^2$ , for every integer  $n> 2$ . Determine the value of  $a_3 + a_5$ .


If  $x$  is a real number, let  $\lfloor x \rfloor$  be the greatest integer that is less than or equal to  $x$ .  If  $n$  is a positive integer, let  $S(n)$  be defined by
\[
  S(n)
    = \left\lfloor \frac{n}{10^{\lfloor \log n \rfloor}} \right\rfloor
        + 10 \left( n - 10^{\lfloor \log n \rfloor}
               \cdot \left\lfloor \frac{n}{10^{\lfloor \log n \rfloor}} \right\rfloor
             \right) \, .
\]
(All the logarithms are base 10.)  How many integers  $n$  from 1 to 2011 (inclusive) satisfy  $S(S(n)) = n$ ?


Four rays spread out from point  $O$  in a  $3$ -dimensional space in a way that the angle between every two rays is  $a$ . Find  $\cos a$ .


Find all triples  $(a,b,c)$  of positive integers such that the product of any two of them when divided by the third leaves the remainder  $1$ .

Solution: \boxed{(2, 3, 5)}


Find all  polynomials  $P(x)$  with real coefficents such that \[P(P(x))=(x^2+x+1)\cdot P(x)\]   where  $x \in \mathbb{R}$

Solution: \boxed{P(x) = 0 \text{ or } P(x) = x^2 + x}


Let  $1\le k\le n$  be integers. At most how many  $k$ -element subsets can we select from  $\{1,2,\dots,n\}$  such that for any two selected subsets, one of the subsets consists of the  $k$  smallest elements of their union?


$p, q, r$  are distinct prime numbers which satisfy  $$ 2pqr + 50pq = 7pqr + 55pr = 8pqr + 12qr = A $$  for natural number  $A$ . Find all values of  $A$ .


An $8$-foot by $10$-foot floor is tiled with square tiles of size $1$ foot by $1$ foot. Each tile has a pattern consisting of four white quarter circles of radius $\frac{1}{2}$ foot centered at each corner of the tile. The remaining portion of the tile is shaded. Find the area of the shaded portion of the floor.

Solution: \boxed{80 - 20\pi}


A quadrilateral  $ABCD$  has a right angle at  $\angle ABC$  and satisfies  $AB = 12$ ,  $BC = 9$ ,  $CD = 20$ , and  $DA = 25$ . Determine  $BD^2$ .
.


Volume  $A$  equals one fourth of the sum of the volumes  $B$  and  $C$ , while volume  $B$  equals one sixth of the sum of the volumes  $A$  and  $C$ . There are relatively prime positive integers  $m$  and  $n$  so that the ratio of volume  $C$  to the sum of the other two volumes is  $\frac{m}{n}$ . Find  $m+n$ .


On a blackboard lies  $50$  magnets in a line numbered from  $1$  to  $50$ , with different magnets containing different numbers.  David walks up to the blackboard and rearranges the magnets into some arbitrary order.  He then writes underneath each pair of consecutive magnets the positive difference between the numbers on the magnets.  If the expected number of times he writes the number  $1$  can be written in the form  $\tfrac mn$  for relatively prime positive integers  $m$  and  $n$ , compute  $100m+n$ .

*Proposed by David Altizio*


For any real numbers  $x,y$  that satisfies the equation   $$ x+y-xy=155 $$   and   $$ x^2+y^2=325 $$  , Find  $|x^3-y^3|$


Given the function   \[ \frac{1}{\left|x+1\right|+\left|x+2\right|+\left|x-3\right|}, \] determine the maximum value of this function.


Find all  $3$ -digit numbers  $\overline{abc}$  ( $a,b \ne 0$ ) such that  $\overline{bcd} \times  a = \overline{1a4d}$  for some integer  $d$  from  $1$  to  $9$


In triangle $ABC$ lines $CE$ and $AD$ are drawn so that $\dfrac{CD}{DB}=\dfrac{3}{1}$ and $\dfrac{AE}{EB}=\dfrac{3}{2}$. Let $r=\dfrac{CP}{PE}$ where $P$ is the intersection point of $CE$ and $AD$. Then $r$ equals $\dfrac{CP}{PE}$.


Find all the real numbers  $a$  and  $b$  that satisfy the relation  $2(a^2 + 1)(b^2 + 1) = (a + 1)(b + 1)(ab + 1)$

Solution: \boxed{(1, 1) \text{ or } (-1, -1)}


Vasya has  $n{}$  candies of several types, where  $n>145$ . It is known that for any group of at least 145 candies, there is a type of candy which appears exactly 10 times. Find the largest possible value of  $n{}$ .

*Proposed by A. Antropov*


In triangle $ABC$ with $\overline{AB}=\overline{AC}=3.6$, a point $D$ is taken on $AB$ at a distance $1.2$ from $A$. Point $D$ is joined to $E$ in the prolongation of $AC$ so that triangle $AED$ is equal in area to $ABC$. Calculate the length of $\overline{AE}$.


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


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


There are  $n$  students standing in line positions  $1$  to  $n$ . While the teacher looks away, some students change their positions. When the teacher looks back, they are standing in line again. If a student who was initially in position  $i$  is now in position  $j$ , we say the student moved for  $|i-j|$  steps. Determine the maximal sum of steps of all students that they can achieve.

Solution: \boxed{\left\lfloor \frac{n^2}{2} \right\rfloor}


What is the greatest integer  $n$  such that   $n\leq1+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\cdots+\frac{1}{\sqrt{2014}}$


Suppose  $28$  objects are placed along a circle at equal distances. In how many ways can  $3$  objects be chosen from among them so that no two of the three chosen objects are adjacent nor diametrically opposite?


Let  $p$  be a real number and  $c\neq 0$  such that \[c-0.1<x^p\left(\dfrac{1-(1+x)^{10}}{1+(1+x)^{10}}\right)<c+0.1\] for all (positive) real numbers  $x$  with  $0<x<10^{-100}$ .  (The exact value  $10^{-100}$  is not important.  You could replace it with any "sufficiently small number".)

Find the ordered pair  $(p,c)$ .

Solution: \boxed{(-1, -5)}


Find the sum of the real roots of the equation $x^4-4x^3+5x^2-4x+1 = 0$.


Given a real number $a \in (1,9)$, find the number of values of $a$ for which the expression $a - \frac{1}{a}$ is an integer.


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

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


Find the sphere of maximal radius that can be placed inside every tetrahedron that has all altitudes of length greater than or equal to  $1.$


Let  $n\geq 2$  be a given integer. Initially, we write  $n$  sets on the blackboard and do a sequence of moves as follows: choose two sets  $A$  and  $B$  on the blackboard such that none of them is a subset of the other, and replace  $A$  and  $B$  by  $A\cap B$  and  $A\cup B$ . This is called a  $\textit{move}$ .
Find the maximum number of moves in a sequence for all possible initial sets.


Given that $p$ and $q$ are positive integers such that $\frac{5}{9} < \frac{p}{q} < \frac{4}{7}$ and $q$ is as small as possible, find $q-p$.


Given $f:\mathbb{R} \rightarrow \mathbb{R}$ satisfies $f(x)f(y)f(z)=12f(xyz)-16xyz$ for every real $x,y,z$, how many such functions f exist?


$P(x)$  is a polynomial with real coefficients such that  $P(a_1) = 0, P(a_{i+1}) = a_i$  ( $i = 1, 2,\ldots$ ) where  $\{a_i\}_{i=1,2,\ldots}$  is an infinite sequence of distinct natural numbers. Determine the possible values of degree of  $P(x)$ .

Solution: \boxed{\deg P = 1}


Let  $a, b$ , and  $c$  be integers that satisfy  $2a + 3b = 52$ ,  $3b + c = 41$ , and  $bc = 60$ . Find  $a + b + c$


In  $\triangle ABC$ ,  $\angle A = 30^{\circ}$  and  $AB = AC = 16$  in. Let  $D$  lie on segment  $BC$  such that  $\frac{DB}{DC} = \frac23$  . Let  $E$  and  $F$  be the orthogonal projections of  $D$  onto  $AB$  and  $AC$ , respectively. Find  $DE + DF$  in inches.


Determine all primes  $p$ , for which there exist positive integers  $m, n$ , such that  $p=m^2+n^2$  and  $p|m^3+n^3+8mn$ .


Determine the general term of the sequence ( $a_n$ ) given by  $a_0 =\alpha > 0$  and  $a_{n+1} =\frac{a_n}{1+a_n}$ .

Solution: \boxed{a_n = \frac{\alpha}{1 + \alpha n}}


Let  $ABC$ be an isosceles triangle with  $\left|AB\right| \equal{} \left|AC\right| \equal{} 10$  and  $\left|BC\right| \equal{} 12$.  $P$ and $R$ are points on $\left[BC\right]$ such that $\left|BP\right| \equal{} \left|RC\right| \equal{} 3$.  $S$ and $T$ are midpoints of $\left[AB\right]$ and $\left[AC\right]$, respectively. If $M$ and $N$ are the foot of perpendiculars from $S$ and $R$ to $PT$, then find $\left|MN\right|$.


Given eight distinguishable rings, let  $n$  be the number of possible five-ring arrangements on the four fingers (not the thumb) of one hand.  The order of rings on each finger is significant, but it is not required that each finger have a ring.  Find the leftmost three nonzero digits of  $n.$


Let  $x,y$  be real numbers such that  $xy=1$ . Let  $T$  and  $t$  be the largest and smallest values of the expression  $\hspace{2cm} \frac{(x+y)^2-(x-y)-2}{(x+y)^2+(x-y)-2}$ .

If  $T+t$  can be expressed in the form  $\frac{m}{n}$  where  $m,n$  are nonzero integers with  $GCD(m,n)=1$ , find the value of  $m+n$ .


Find the total number of primes  $p<100$  such that  $\lfloor (2+\sqrt{5})^p \rfloor-2^{p+1}$  is divisible by  $p$ . Here  $\lfloor x \rfloor$  denotes the greatest integer less than or equal to  $x$ .


Given a positive real number A that rounds to 20 and another positive real number B that rounds to 23, find the largest possible value of the largest integer not exceeding the value of $\frac{100A}{B}$.


Adi the Baller is shooting hoops, and makes a shot with probability  $p$ . He keeps shooting hoops until he misses. The value of  $p$  that maximizes the chance that he makes between 35 and 69 (inclusive) buckets can be expressed as  $\frac{1}{\sqrt[b]{a}}$  for a prime  $a$  and positive integer  $b$ . Find  $a+b$ .

Proposed by Minseok Eli Park (wolfpack)


Given a square  $ABCD$  whose side length is  $1$ ,  $P$  and  $Q$  are points on the sides  $AB$  and  $AD$ . If the perimeter of  $APQ$  is  $2$  find the angle  $PCQ$ .


$\textbf{Problem 2.}$  The list of all the numbers of  $5$  different numbers that are formed with the digits  $1,2,3,4$  and  $5$  is made.
In this list the numbers are ordered from least to greatest. Find the number that occupies the  $100th$  position in the list.


Find the maximum positive integer  $k$  such that for any positive integers  $m,n$  such that  $m^3+n^3>(m+n)^2$ , we have  $$ m^3+n^3\geq (m+n)^2+k $$  *Proposed by Dorlir Ahmeti, Albania*


A cuboid has an integer volume. Three of the faces have different areas, namely  $7, 27$ , and  $L$ . What is the smallest possible integer value for  $L$ ?


Find all strictly increasing sequences of positive integers  $a_1, a_2, \ldots$  with  $a_1=1$ , satisfying   $$ 3(a_1+a_2+\ldots+a_n)=a_{n+1}+\ldots+a_{2n} $$   for all positive integers  $n$ .

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


What is the smallest positive integer  $k$  such that  $k(3^3 + 4^3 + 5^3) = a^n$  for some positive integers  $a$  and  $n$ , with  $n > 1$ ?


Let  $P(x)=x^3+ax^2+bx+c$  be a polynomial where  $a,b,c$  are integers and  $c$   is odd. Let  $p_{i}$  be the value of  $P(x)$  at  $x=i$ . Given that  $p_{1}^3+p_{2}^{3}+p_{3}^{3}=3p_{1}p_{2}p_{3}$ , find the value of  $p_{2}+2p_{1}-3p_{0}.$


$1989$  equal circles are arbitrarily placed on the table without overlap. What is the least number of colors are needed such that all the circles can be painted with any two tangential circles colored differently.


What is the volume of the region in three-dimensional space defined by the inequalities  $|x|+|y|+|z|\le1$  and  $|x|+|y|+|z-1|\le1$ ?


Determine all pairs of *distinct* real numbers  $(x, y)$  such that both of the following are true:

- $x^{100} - y^{100} = 2^{99} (x-y)$
- $x^{200} - y^{200} = 2^{199} (x-y)$

Solution: \boxed{(2, 0) \text{ and } (0, 2)}


Find the sum of all positive integers  $1\leq k\leq 99$  such that there exist positive integers  $a$  and  $b$  with the property that \[x^{100}-ax^k+b=(x^2-2x+1)P(x)\] for some polynomial  $P$  with integer coefficients.

*Proposed by David Altizio*


Let  $ABC$  be a triangle with  $AC\neq BC$ , and let  $A^{\prime }B^{\prime }C$  be a triangle obtained from  $ABC$  after some rotation centered at  $C$ . Let  $M,E,F$  be the midpoints of the segments  $BA^{\prime },AC$  and  $CB^{\prime }$   respectively. If  $EM=FM$ , find  $\widehat{EMF}$ .


Given the system of equations $x^2 + x^2y^2 + x^2y^4 = 525$ and $x + xy + xy^2 = 35$, find the sum of the real $y$ values that satisfy the equations.


Lynnelle and Moor love toy cars, and together, they have  $27$  red cars,  $27$  purple cars, and  $27$  green cars. The number of red cars Lynnelle has individually is the same as the number of green cars Moor has individually. In addition, Lynnelle has  $17$  more cars of any color than Moor has of any color. How many purple cars does Lynnelle have?


When  $4444^{4444}$  is written in decimal notation, the sum of its digits is  $ A.$  Let  $B$  be the sum of the digits of  $A.$  Find the sum of the digits of  $ B.$  ( $A$  and  $B$  are written in decimal notation.)


Determine all triples  $(x, y, z)$  of nonnegative real numbers that verify the following system of equations:  $$ x^2 - y = (z -1)^2  $$     $$ y^2 - z = (x -1)^2 $$    $$ z^2 - x = (y - 1)^2 $$

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


How many integers  $n$  with  $0\leq n < 840$  are there such that  $840$  divides  $n^8-n^4+n-1$ ?


One night, over dinner Jerry poses a challenge to his younger children: "Suppose we travel  $50$  miles per hour while heading to our final vacation destination..."

Hannah teases her husband, "You  $\textit{would}$  drive that  $\textit{slowly}\text{!}$ "

Jerry smirks at Hannah, then starts over, "So that we get a good view of all the beautiful landscape your mother likes to photograph from the passenger's seat, we travel at a constant rate of  $50$  miles per hour on the way to the beach.  However, on the way back we travel at a faster constant rate along the exact same route.  If our faster return rate is an integer number of miles per hour, and our average speed for the  $\textit{whole round trip}$  is  $\textit{also}$  an integer number of miles per hour, what must be our speed during the return trip?"

Michael pipes up, "How about  $4950$  miles per hour?!"

Wendy smiles, "For the sake of your  $\textit{other}$  children, please don't let  $\textit{Michael}$  drive."

Jerry adds, "How about we assume that we never  $\textit{ever}$  drive more than  $100$  miles per hour.  Michael and Wendy, let Josh and Alexis try this one."

Joshua ignores the problem in favor of the huge pile of mashed potatoes on his plate.  But Alexis scribbles some work on her napkin and declares the correct answer.  What answer did Alexis find?


Consider a circle of radius  $4$  with center  $O_1$ , a circle of radius  $2$  with center  $O_2$  that lies on the circumference of circle  $O_1$ , and a circle of radius  $1$  with center  $O_3$  that lies on the circumference of circle  $O_2$ . The centers of the circle are collinear in the order  $O_1$ ,  $O_2$ ,  $O_3$ . Let  $A$  be a point of intersection of circles  $O_1$  and  $O_2$  and  $B$  be a point of intersection of circles  $O_2$  and  $O_3$  such that  $A$  and  $B$  lie on the same semicircle of  $O_2$ . Compute the length of  $AB$ .


Given $a$ and $b$ are the roots of the equation $x^2-2cx-5d = 0$, and $c$ and $d$ are the roots of the equation $x^2-2ax-5b=0$, where $a,b,c,d$ are distinct real numbers, find $a+b+c+d$.


When  $126$  is added to its reversal,  $621,$  the sum is  $126 + 621 = 747.$  Find the greatest integer which when added to its reversal yields  $1211.$


Let  $x$  and  $y$  be real numbers with  $x>y$  such that  $x^2y^2+x^2+y^2+2xy=40$  and  $xy+x+y=8$ .  Find the value of  $x$ .

Solution: \boxed{3 + \sqrt{7}}


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


A cube with side length  $100cm$  is filled with water and has a hole through which the water drains into a cylinder of radius  $100cm.$  If the water level in the cube is falling at a rate of  $1 \frac{cm}{s} ,$  how fast is the water level in the cylinder rising?


Determine all  $n \geq 3$  for which there are  $n$  positive integers  $a_1, \cdots , a_n$  any two of which have a common divisor greater than  $1$ , but any three of which are coprime. Assuming that, moreover, the numbers  $a_i$  are less than  $5000$ , find the greatest possible  $n$ .

Solution: \boxed{n = 4}


The repeating decimal  $2.0151515\ldots$  can be expressed as  $\tfrac{m}{n}$ , where  $m$  and  $n$  are relatively prime positive integers. Find  $m + n$ .


Consider a right-angled triangle  $ABC$  with  $\angle C = 90^o$ . Suppose that the hypotenuse  $AB$  is divided into four equal parts by the points  $D,E,F$ , such that  $AD = DE = EF = FB$ . If  $CD^2 +CE^2 +CF^2 = 350$ , find the length of  $AB$ .


Find the largest integer  $n$  such that  $n$  is divisible by all positive integers less than  $\sqrt[3]{n}$ .


Triangle  $ ABC$  obeys  $ AB = 2AC$  and  $ \angle{BAC} = 120^{\circ}.$  Points  $ P$  and  $ Q$  lie on segment  $ BC$  such that
\begin{eqnarray*}
AB^2 + BC \cdot CP = BC^2
3AC^2 + 2BC \cdot CQ = BC^2
\end{eqnarray*}
Find  $ \angle{PAQ}$  in degrees.


In a triangle  $ ABC$  with  $ AB\equal{}20, AC\equal{}21$  and  $ BC\equal{}29$ , points  $ D$  and  $ E$  are taken on the segment  $ BC$  such that  $ BD\equal{}8$  and  $ EC\equal{}9$ . Calculate the angle  $ \angle DAE$ .


1. Baron Munchhausen was told that some polynomial  $P(x)=a_{n} x^{n}+\ldots+a_{1} x+a_{0}$  is such that  $P(x)+P(-x)$  has exactly 45 distinct real roots. Baron doesn't know the value of  $n$ . Nevertheless he claims that he can determine one of the coefficients  $a_{n}, \ldots, a_{1}, a_{0}$  (indicating its position and value). Isn't Baron mistaken?

	Boris Frenkin

Solution: \boxed{a_0 = 0}


Let  $a$  be a real number. Suppose the function  $f(x) = \frac{a}{x-1} + \frac{1}{x-2} + \frac{1}{x-6}$  defined in the interval  $3 < x < 5$  attains its maximum at  $x=4$ . Find the value of  $a.$


Find all pairs of positive integers  $(x,y) $  for which  $x^3 + y^3 = 4(x^2y + xy^2 - 5) .$

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


Let $x=\cos 36^{\circ} - \cos 72^{\circ}$. Simplify the expression for $x$.


Given a ball of mass  $m$  is launched into the air, ignoring air resistance and experiencing a constant wind force  $F_0$  in the - $x$  direction, determine the angle above the positive  $x$ -axis at which it must be launched to return to its initial point.

Solution: \boxed{\theta = \tan^{-1} \left( \frac{mg}{F_0} \right)}


In a right triangle, the ratio of the hypotenuse to the perimeter of the triangle determines an interval on the real numbers. Find the midpoint of this interval.

Solution: \boxed{\frac{2\sqrt{2} - 1}{4}}


let  $x,y,z$  be  positive reals , such that  $x+y+z=1399$  find the
   $$ \max( [x]y + [y]z + [z]x )  $$
(  $[a]$  is the biggest integer not exceeding  $a$ )

Solution: \boxed{\max( \lfloor x \rfloor y + \lfloor y \rfloor z + \lfloor z \rfloor x ) = 652400}


Two sums, each consisting of  $n$  addends , are shown below: $S = 1 + 2 + 3 + 4 + ...$  $T = 100 + 98 + 96 + 94 +...$  .
For what value of  $n$  is it true that  $S = T$  ?


The polynomials  $P_{n}(x)$  are defined by  $P_{0}(x)=0,P_{1}(x)=x$  and \[P_{n}(x)=xP_{n-1}(x)+(1-x)P_{n-2}(x) \quad n\geq 2\] For every natural number  $n\geq 1$ , find all real numbers  $x$  satisfying the equation  $P_{n}(x)=0$ .

Solution: \boxed{x = 0}


Lazim rolls two  $24$ -sided dice. From the two rolls, Lazim selects the die with the highest number.  $N$  is an integer not greater than  $24$ . What is the largest possible value for  $N$  such that there is a more than  $50$ % chance that the die Lazim selects is larger than or equal to  $N$ ?


Find all positive integer pairs  $(a, b)$  such that  $\frac{2^{a+b}+1}{2^a+2^b+1}$  is an integer.**General**
Find all positive integer triples  $(t, a, b)$  such that  $\frac{t^{a+b}+1}{t^a+t^b+1}$  is an integer.

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


Let $n$ and $m$ be integers such that $n\leq 2007 \leq m$ and $n^n \equiv -1 \equiv m^m \pmod 5$. What is the least possible value of $m-n$?


Let  $x$  be the smallest positive integer for which  $2x$  is the square of an integer,  $3x$  is the third power of an integer, and  $5x$  is the fifth power of an integer. Find the prime factorization of  $x$ .

(St. Wagner, Stellenbosch University)

Solution: \boxed{2^{15} \cdot 3^{20} \cdot 5^{24}}


In a coordinate system, a circle with radius  $7$  and center is on the y-axis placed inside the parabola with equation  $y = x^2$  , so that it just touches the parabola in two points. Determine the coordinate set for the center of the circle.

Solution: \boxed{(0, \frac{197}{4})}


Let  $f(x) = x^3 + 3x - 1$  have roots  $ a, b, c$ . Given that  $\frac{1}{a^3 + b^3}+\frac{1}{b^3 + c^3}+\frac{1}{c^3 + a^3}$  can be written as  $\frac{m}{n}$ , where  $m, n$  are positive integers and  $gcd(m, n) = 1$ , find  $100m + n$ .


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}


The sum of the squares of three positive numbers is  $160$ . One of the numbers is equal to the sum of the other two. The difference between the smaller two numbers is  $4.$  What is the difference between the cubes of the smaller two numbers?

*Author: Ray Li*

ClarificationThe problem should ask for the positive difference.


Determine the largest possible value of the expression  $ab+bc+ 2ac$  for non-negative real numbers  $a, b, c$  whose sum is  $1$ .


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}


Given the linear fractional transformation of  $x$  into  $f_1(x)=\dfrac{2x-1}{x+1}$ , define  $f_{n+1}(x)=f_1(f_n(x))$  for  $n=1,2,3,\cdots$ . Express  $f_{28}(x)$  in terms of  $x$ .


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.


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}


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


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


Given an isolated ground state iron atom with atomic number Z = 26, determine the number of orbitals that contain one or more electrons.


Find the smallest volume bounded by the coordinate planes and  by a tangent plane to the ellipsoid  $$ \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}=1. $$

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


Suppose  $a,b,c,x,y,z$  are pairwisely different real numbers. How many terms in the following can be  $1$  at most:   $$ \begin{aligned}
&ax+by+cz,&&&&ax+bz+cy,&&&&ay+bx+cz,
&ay+bz+cx,&&&&az+bx+cy,&&&&az+by+cx?
\end{aligned} $$


Lunasa, Merlin, and Lyrica each has an instrument. We know the following about the prices of their instruments:

(a) If we raise the price of Lunasa's violin by  $50\%$  and decrease the price of Merlin's trumpet by  $50\%$ , the violin will be  $\$ 50 $ more expensive than the trumpet;
(b) If we raise the price of Merlin's trumpet by $ 50\% $ and decrease the price of Lyrica's piano by $ 50\% $, the trumpet will be $ \ $50$  more expensive than the piano.

Given these conditions only, there exist integers  $m$  and  $n$  such that if we raise the price of Lunasa's violin by  $m\%$  and decrease the price of Lyrica's piano by  $m\%$ , the violin must be exactly  $\$ n $ more expensive than the piano. Find $ 100m+n$.

*Proposed by Yannick Yao*


A broken line consists of  $31$  segments. It has no self intersections, and its start and end points are distinct. All segments are extended to become straight lines. Find the least possible number of straight lines.


Ada rolls a standard  $4$ -sided die  $5$  times. The probability that the die lands on at most two distinct sides can be written as  $ \frac{A}{B}$  for relatively prime positive integers  $A$  and  $B$ . Find  $1000A +B$


If the inequality $((x+y)^2+4)((x+y)^2-2)\geq A\cdot (x-y)^2$ holds for every real numbers $x,y$ such that $xy=1$, determine the largest value of $A$.


Given a cube, calculate the number of rectangles that can be formed by its vertices.


What quadratic polynomial whose coeﬃcient of  $x^2$  is  $1$  has roots which are the complex conjugates of the solutions of  $x^2 -6x+ 11 = 2xi-10i$ ? (Note that the complex conjugate of  $a+bi$  is  $a-bi$ , where a and b are real numbers.)

Solution: \boxed{x^2 - (6 - 2i)x + (11 - 10i)}


Let  $(a_n)_{n \geq 0}$  be the sequence of integers defined recursively by  $a_0 = 0, a_1 = 1, a_{n+2} = 4a_{n+1} + a_n$  for  $n \geq 0.$  Find the common divisors of  $a_{1986}$  and  $a_{6891}.$


Find all nonzero subsets  $A$  of the set  $\left\{2,3,4,5,\cdots\right\}$  such that  $\forall n\in A$ , we have that  $n^{2}+4$  and  $\left\lfloor{\sqrt{n}\right\rfloor}+1$  are both in  $A$ .

Solution: \boxed{\{2, 3, 4, 5, \ldots\}}


Let $ABC$ be a triangle such that $|AB|=8$ and $|AC|=2|BC|$. What is the largest value of altitude from side $[AB]$?


Three fair six-sided dice are thrown. Determine the probability that the sum of the numbers on the three top faces is  $6$ .


Let the sequence  $a_{1}, a_{2}, \cdots$  be defined recursively as follows:  $a_{n}=11a_{n-1}-n$ . If all terms of the sequence are positive, the smallest possible value of  $a_{1}$  can be written as  $\frac{m}{n}$ , where  $m$  and  $n$  are relatively prime positive integers. What is  $m+n$ ?


Find all functions  $\displaystyle f : \mathbb N^\ast \to \mathbb N^\ast$  ( $\displaystyle N^\ast = \{ 1,2,3,\ldots \}$ ) with the property that, for all  $\displaystyle n \geq 1$ , \[ f(1) + f(2) + \ldots + f(n)  \] is a perfect cube  $\leq n^3$ .

*Dinu Teodorescu*

Solution: \boxed{f(n) = 3n^2 - 3n + 1}


For what digit A is the numeral 1AA a perfect square in base-5 and a perfect cube in base-6?


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


Six people each flip a fair coin. Everyone who flipped tails then flips their coin again. Given that the probability that all the coins are now heads can be expressed as simplified fraction  $\tfrac{m}{n}$ , compute  $m+n$ .


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

(M.Kungozhin)


During a break,  $n$  children at school sit in a circle around their teacher to play a game.  The teacher walks clockwise close to the children and hands out candies to some of them according to the following rule:

He selects one child and gives him a candy, then he skips the next child and gives a candy to the next one, then he skips 2 and gives a candy to the next one, then he skips 3, and so on.

Determine the values of  $n$  for which eventually, perhaps after many rounds, all children will have at least one candy each.

Solution: \boxed{n = 2^k}


Find all the integers  $x, y$  and  $z$  greater than or equal to  $0$  such that  $2^x + 9 \cdot  7^y = z^3$

Solution: \boxed{(0, 1, 4)}


There are  $9$  cards with the numbers  $1, 2, 3, 4, 5, 6, 7, 8$  and  $9$ . What is the largest number of these cards can be decomposed in a certain order in a row, so that in any two adjacent cards, one of the numbers is divided by the other?


**<u>BdMO National 2015 Secondary Problem 1.</u>**

A crime is committed during the hartal.There are four witnesses.The witnesses are logicians and make the following statement:
Witness **One**    said  exactly one  of the four witnesses is a liar.

Witness  **Two**  said  exactly  two  of the four witnesses is a liar.

Witness **Three**    said  exactly three of the four witnesses is a liar.

Witness  **Four**  said  exactly four  of the four witnesses is a liar.

Assume that each of the statements is either true or false.How many of the winesses are liars?


$(x_{n})_{-\infty<n<\infty}$  is a sequence of real numbers which satisfies  $x_{n+1}=\frac{x_{n}^2+10}{7}$  for every  $n \in \mathbb{Z}$ . If there exist a real upperbound for this sequence, find all the values  $x_{0}$  can take.

Solution: \boxed{[2, 5]}


Solution: \boxed{\frac{1 + 5^{21}}{6}}


What is the greatest integer not exceeding the number $\left( 1 + \frac{\sqrt 2 + \sqrt 3 + \sqrt 4}{\sqrt 2 + \sqrt 3 + \sqrt 6 + \sqrt 8 + 4}\right)^{10}$?


The triangle  $K_2$  has as its vertices the feet of the altitudes of a non-right triangle  $K_1$ . Find all possibilities for the sizes of the angles of  $K_1$  for which the triangles  $K_1$  and  $K_2$  are similar.

Solution: \boxed{A = 60^\circ, B = 60^\circ, C = 60^\circ}


An infinite sequence of positive real numbers is defined by  $a_0=1$  and  $a_{n+2}=6a_n-a_{n+1}$  for  $n=0,1,2,\cdots$ .  Find the possible value(s) of  $a_{2007}$ .


Given $x^2 - x - 1$ is a factor of $ax^3 + bx^2 + 1$, find the value of $b$.


For how many integers  $n$ does the equation system
\[\begin{array}{rcl}
2x+3y &=& 7 \\
5x + ny &=& n^2
\end{array}\]
have a solution over integers.


Let  $n$  be a positive integer. We call  $(a_1,a_2,\cdots,a_n)$  a *good*  $n-$ tuple if  $\sum_{i=1}^{n}{a_i}=2n$  and there doesn't exist a set of  $a_i$ s such that the sum of them is equal to  $n$ . Find all *good*  $n-$ tuple.
(For instance,  $(1,1,4)$  is a *good*  $3-$ tuple, but  $(1,2,1,2,4)$  is not a *good*  $5-$ tuple.)

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


Find all natural numbers  $n$  such that the equation  $x^2 + y^2 + z^2 = nxyz$  has solutions in positive integers

Solution: \boxed{n = 3}


Let  $X = \{-5,-4,-3,-2,-1,0,1,2,3,4,5\}$  and  $S = \{(a,b)\in X\times X:x^2+ax+b \text{ and  }x^3+bx+a \text{ have at least a common real zero .}\}$ How many elements are there in  $S$ ?


A club with $x$ members is organized into four committees in accordance with these two rules: each member belongs to two and only two committees, and each pair of committees has one and only one member in common. Determine the possible values of $x$.


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


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

What is the largest number of the marbels you have?


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


Let  $a,b\in\mathbb{R},~a>1,~b>0.$  Find the least possible value for  $\alpha$  such that :  $$ (a+b)^x\geq a^x+b,~(\forall)x\geq\alpha. $$


Find, with proof, a polynomial  $f(x,y,z)$  in three variables, with integer coefficients, such that for all  $a,b,c$  the sign of  $f(a,b,c)$  (that is, positive, negative, or zero) is the same as the sign of  $a+b\sqrt[3]{2}+c\sqrt[3]{4}$ .

Solution: \boxed{f(x, y, z) = x^3 + 2y^3 + 4z^3 - 6xyz}


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


$f$  is a function on the set of complex numbers such that  $f(z)=1/(z*)$ , where  $z*$  is the complex conjugate of  $z$ .  $S$  is the set of complex numbers  $z$  such that the real part of  $f(z)$  lies between  $1/2020$  and  $1/2018$ . If  $S$  is treated as a subset of the complex plane, the area of  $S$  can be expressed as  $m× \pi$  where  $m$  is an integer. What is the value of  $m$ ?


Among all polynomials  $P(x)$  with integer coefficients for which  $P(-10) = 145$  and  $P(9) = 164$ , compute the smallest possible value of  $|P(0)|.$


The perpendicular bisectors of the sides  $AB$  and  $CD$  of the rhombus  $ABCD$  are drawn. It turned out that they divided the diagonal  $AC$  into three equal parts. Find the altitude of the rhombus if  $AB = 1$ .


Given that  $(1+\sin t)(1+\cos t)=5/4$  and \[ (1-\sin t)(1-\cos t)=\frac mn-\sqrt{k}, \] where  $k, m,$  and  $n$  are positive integers with  $m$  and  $n$  relatively prime, find  $k+m+n.$


Let  $ x,y$  are  real numbers such that  $x^2+2cosy=1$ . Find the ranges of  $x-cosy$ .

Solution: \boxed{[-1, 1 + \sqrt{3}]}


If  $a$  and  $b$  are the roots of  $x^2 - 2x + 5$ , what is  $|a^8 + b^8|$ ?


Given the equation $x = |2x - |60 - 2x||$, find the sum of all the solutions.


Find the positive integer  $k$  such that the roots of  $x^3 - 15x^2 + kx -1105$  are three distinct collinear points in the complex plane.


Find all possible values of integer  $n > 3$  such that there is a convex  $n$ -gon in which, each diagonal is the perpendicular bisector of at least one other diagonal.

Proposed by Mahdi Etesamifard

Solution: \boxed{n = 4}


There are 2012 backgammon checkers with one side being black and the other side being white. These checkers are arranged in a line such that no two consecutive checkers are the same color. At each move, two checkers are chosen, and their colors are reversed along with those of the checkers between them. Determine the minimum number of moves required to make all checkers the same color.


Let  $ABCD$  be a rectangle in which  $AB + BC + CD = 20$  and  $AE = 9$  where  $E$  is the midpoint of the side  $BC$ . Find the area of the rectangle.


Let $f(n)$ be the smallest prime which divides $n^4+1$. Determine the remainder when the sum $f(1)+f(2)+\cdots+f(2014)$ is divided by 8$.


A circle with a radius of $16$ feet has a rhombus formed by two radii and two chords. Calculate the area of the rhombus.


Mr. Squash bought a large parking lot in Utah, which has an area of  $600$  square meters. A car needs  $6$  square meters of parking space while a bus needs  $30$  square meters of parking space. Mr. Squash charges  $\$ 2.50 $ per car and $ \ $7.50$  per bus, but Mr. Squash can only handle at most  $60$  vehicles at a time. Find the ordered pair  $(a,b)$  where  $a$  is the number of cars and  $b$  is the number of buses that maximizes the amount of money Mr. Squash makes.

*Proposed by Nathan Cho*

Solution: \boxed{(50, 10)}


Find the smallest positive integer  $n$  such that the  $73$  fractions  $\frac{19}{n+21}, \frac{20}{n+22},\frac{21}{n+23},...,\frac{91}{n+93}$  are all irreducible.


Determine all intergers  $n\geq 2$  such that  $a+\sqrt{2}$  and  $a^n+\sqrt{2}$  are both rational for some real number  $a$  depending on  $n$

Solution: \boxed{n = 2}


For any finite set  $S$ , let  $f(S)$  be the sum of the elements of  $S$  (if  $S$  is empty then  $f(S)=0$ ).  Find the sum over all subsets  $E$  of  $S$  of  $\dfrac{f(E)}{f(S)}$  for  $S=\{1,2,\cdots,1999\}$ .


If  $\phi$  is the Golden Ratio, we know that  $\frac1\phi = \phi - 1$ .  Define a new positive real number, called  $\phi_d$ , where  $\frac1{\phi_d} = \phi_d - d$  (so  $\phi = \phi_1$ ).  Given that  $\phi_{2009} = \frac{a + \sqrt{b}}{c}$ ,  $a, b, c$  positive integers, and the greatest common divisor of  $a$  and  $c$  is 1, find  $a + b + c$ .


$k$  marbles are placed onto the cells of a  $2024 \times 2024$  grid such that each cell has at most one marble and there are no two marbles are placed onto two neighboring cells (neighboring cells are defined as cells having an edge in common).

a) Assume that  $k=2024$ . Find a way to place the marbles satisfying the conditions above, such that moving any placed marble to any of its neighboring cells will give an arrangement that does not satisfy both the conditions.
b) Determine the largest value of  $k$  such that for all arrangements of  $k$  marbles satisfying the conditions above, we can move one of the placed marble onto one of its neighboring cells and the new arrangement satisfies the conditions above.

Solution: \boxed{k = 2023}


Choose a permutation of $ \{1,2, ..., 20\}$  at random. Let  $m$  be the amount of numbers in the permutation larger than all numbers before it. Find the expected value of  $2^m$ .

*Proposed by Evan Chang (squareman), USA*


Given that square $ABCD$ is divided into four rectangles by $EF$ and $GH$, where $EF$ is parallel to $AB$ and $GH$ is parallel to $BC$, $\angle BAF = 18^\circ$, $EF$ and $GH$ meet at point $P$, and the area of rectangle $PFCH$ is twice that of rectangle $AGPE$, find the value of $\angle FAH$ in degrees.


Calculate the following indefinite integral.

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

[2]  $\int \sin x\cos 3x dx$

[3]  $\int \sin 2x\sin 3x dx$

[4]  $\int \frac{dx}{4x^2-12x+9}$

[5]  $\int \cos ^4 x dx$

Solution: \boxed{\frac{3}{8} x + \frac{1}{4} \sin 2x + \frac{1}{32} \sin 4x + C}


Let  $ABC$  be a triangle with  $AC = 28$ ,  $BC = 33$ , and  $\angle ABC = 2\angle ACB$ . Compute the length of side  $AB$ .

*2016 CCA Math Bonanza #10*


For some positive integer  $n$, the number  $110n^3$  has  $110$  positive integer divisors, including  $1$  and the number  $110n^3$. Determine the number of positive integer divisors of the number  $81n^4$.


On a clock, there are two instants between  $12$  noon and  $1 \,\mathrm{PM}$ , when the hour hand and the minute hannd are at right angles. The difference *in minutes* between these two instants is written as  $a + \dfrac{b}{c}$ , where  $a, b, c$  are positive integers, with  $b < c$  and  $b/c$  in the reduced form. What is the value of  $a+b+c$ ?


Find the ordered triple of natural numbers  $(x,y,z)$  such that  $x \le y \le z$  and  $x^x+y^y+z^z = 3382.$ [i]Proposed by Evan Fang

Solution: \boxed{(1, 4, 5)}


Given that there are 5 points in the plane no three of which are collinear, draw all the segments whose vertices are these points, and determine the minimum number of new points made by the intersection of the drawn segments.


Let  $a>1.$  Find the area  $S(a)$  of the part surrounded by the curve  $y=\frac{a^{4}}{\sqrt{(a^{2}-x^{2})^{3}}}\ (0\leq x\leq 1),\ x$  axis ,  $y$  axis and the line  $x=1,$  then  when  $a$  varies in the range of  $a>1,$  then find the extremal value of  $S(a).$


A line parallel to the side  $AC$  of a triangle  $ABC$  with  $\angle C = 90$  intersects side  $AB$  at  $M$  and side  $BC$  at  $N$ , so that  $CN/BN = AC/BC = 2/1$ . The segments  $CM$  and  $AN$  meet at  $O$ . Let  $K$  be a point on the segment  $ON$  such that  $MO+OK = KN$ . The bisector of  $\angle ABC$  meets the line through  $K$  perpendicular to  $AN$  at point  $T$ .
Determine  $\angle MTB$ .

Solution: \boxed{\angle MTB = 90^\circ}


Let  $a, b, c$  be integers not all the same with  $a, b, c\ge 4$  that satisfy   $$ 4abc = (a + 3) (b + 3) (c + 3). $$  Find the numerical value of  $a + b + c$ .


A convex polyhedron has  $m$  triangular faces (there can be faces of other kind too). From each vertex there are exactly 4 edges. Find the least possible value of  $m$ .


Determine the largest natural number  $m$  such that for each non negative real numbers  $a_1 \ge a_2 \ge ... \ge a_{2014} \ge 0$  , it is true that   $$ \frac{a_1+a_2+...+a_m}{m}\ge \sqrt{\frac{a_1^2+a_2^2+...+a_{2014}^2}{2014}} $$

Solution: \boxed{m = 44}


Let  $k$  be a real number such that the product of real roots of the equation   $$ X^4 + 2X^3 + (2 + 2k)X^2 + (1 + 2k)X + 2k = 0 $$   is  $-2013$ . Find the sum of the squares of these real roots.


Given that $\left(1+\sqrt{2}\right)^{2012}=a+b\sqrt{2}$, where $a$ and $b$ are integers, find the greatest common divisor of $b$ and $81$.


There are 22 black and 3 blue balls in a bag. Ahmet chooses an integer  $ n$  in between 1 and 25. Betül draws  $ n$  balls from the bag one by one such that no ball is put back to the bag after it is drawn. If exactly 2 of the  $ n$  balls are blue and the second blue ball is drawn at  $ n^{th}$  order, Ahmet wins, otherwise Betül wins. Calculate the value of  n  that maximizes the chance for Ahmet to win.


For any set  $S$  of five points in the plane, no three of which are collinear, let  $M(S)$  and  $m(S)$  denote the greatest and smallest areas, respectively, of triangles determined by three points from  $S$ .  What is the minimum possible value of  $M(S)/m(S)$  ?

Solution: \boxed{\frac{1 + \sqrt{5}}{2}}


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


If  $a$  and  $b$  are positive integers such that  $a \cdot b = 2400,$  find the least possible value of  $a + b.$


Find the greatest real number  $C$  such that, for all real numbers  $x$  and  $y \neq x$  with  $xy = 2$  it holds that
\[\frac{((x + y)^2 - 6)((x - y)^2 + 8)}{(x-y)^2}\geq C.\]
When does equality occur?


If  $a$  and  $b$  are complex numbers such that  $a^2 + b^2 = 5$  and  $a^3 + b^3 = 7$ , then their sum,  $a + b$ , is real. The greatest possible value for the sum  $a + b$  is  $\tfrac{m+\sqrt{n}}{2}$  where  $m$  and  $n$  are integers. Find  $n.$


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

Solution: \boxed{(2, 1)}


It is possible to arrange eight of the nine numbers  $2, 3, 4, 7, 10, 11, 12, 13, 15$   in the vacant squares of the  $3$  by  $4$  array shown on the right so that the arithmetic average of the numbers in each row and in each column is the same integer. Exhibit such an arrangement, and specify which one of the nine numbers must be left out when completing the array.

[asy]
defaultpen(linewidth(0.7));
for(int x=0;x<=4;++x)
draw((x+.5,.5)--(x+.5,3.5));
for(int x=0;x<=3;++x)
draw((.5,x+.5)--(4.5,x+.5));
label(" $1$ ",(1,3));
label(" $9$ ",(2,2));
label(" $14$ ",(3,1));
label(" $5$ ",(4,2));[/asy]


Find all triples  $(a,b,c)$  of positive integers such that if  $n$  is not divisible by any prime less than  $2014$ , then  $n+c$  divides  $a^n+b^n+n$ .

*Proposed by Evan Chen*

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


Let  $S$  be a real number. It is known that however we choose several numbers from the interval  $(0, 1]$  with sum equal to  $S$ , these numbers can be separated into two subsets with the following property: The sum of the numbers in one of the subsets doesn’t exceed 1 and the sum of the numbers in the other subset doesn’t exceed 5.

Find the greatest possible value of  $S$ .


$ABCD$  is a square of side 1.  $P$  and  $Q$  are points on  $AB$  and  $BC$  such that  $\widehat{PDQ} = 45^{\circ}$ . Find the perimeter of  $\Delta PBQ$ .


A tree has  $10$  pounds of apples at dawn. Every afternoon, a bird comes and eats  $x$  pounds of apples. Overnight, the amount of food on the tree increases by  $10\%$ . What is the maximum value of  $x$  such that the bird can sustain itself indefinitely on the tree without the tree running out of food?


Find all real numbers  $x, y, z$  that satisfy the following system  $$ \sqrt{x^3 - y} = z - 1 $$    $$ \sqrt{y^3 - z} = x - 1 $$    $$ \sqrt{z^3 - x} = y - 1 $$

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


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  $T(x)$


$n$  students take a test with  $m$  questions, where  $m,n\ge 2$  are integers. The score given to every question is as such: for a certain question, if  $x$  students fails to answer it correctly, then those who answer it correctly scores  $x$  points, while those who answer it wrongly scores  $0$ . The score of a student is the sum of his scores for the  $m$  questions. Arrange the scores in descending order  $p_1\ge p_2\ge \ldots \ge p_n$ . Find the maximum value of  $p_1+p_n$ .


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


Determine the largest integer  $n\geq 3$  for which the edges of the complete graph on  $n$  vertices
can be assigned pairwise distinct non-negative integers such that the edges of every triangle have numbers which form an arithmetic progression.

Solution: \boxed{n = 4}


Given the polynomial $a_0x^n+a_1x^{n-1}+\cdots+a_{n-1}x+a_n$, where $n$ is a positive integer or zero, and $a_0$ is a positive integer. The remaining $a$'s are integers or zero. Set $h=n+a_0+|a_1|+|a_2|+\cdots+|a_n|$. Determine the number of polynomials with $h=3$.


For real numbers  $ x_{i}>1,1\leq i\leq n,n\geq 2,$  such that:

 $ \frac{x_{i}^2}{x_{i}\minus{}1}\geq S\equal{}\displaystyle\sum^n_{j\equal{}1}x_{j},$  for all  $ i\equal{}1,2\dots, n$

find, with proof,  $ \sup S.$

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


Find the number of ordered pairs  $(x,y)$  of positive integers that satisfy  $x\le 2y\le 60$  and  $y\le 2x\le 60.$


$2023$  players participated in a tennis tournament, and any two players played exactly one match. There was no draw in any match, and no player won all the other players. If a player  $A$  satisfies the following condition, let  $A$  be "skilled player".**(Condition)** For each player  $B$  who won  $A$ , there is a player  $C$  who won  $B$  and lost to  $A$ .

It turned out there are exactly  $N(\geq 0)$  skilled player. Find the minimum value of  $N$ .


Let  $ ABC$  be a right angled triangle of area 1. Let  $ A'B'C'$  be the points obtained by reflecting  $ A,B,C$  respectively, in their opposite sides. Find the area of  $ \triangle A'B'C'.$


Given that  $ 2^{2004}$  is a  $ 604$ -digit number whose first digit is  $ 1$ , how many elements of the set  $ S \equal{} \{2^0,2^1,2^2, \ldots,2^{2003}\}$  have a first digit of  $ 4$ ?


Ajay is standing at point $A$ near Pontianak, Indonesia, $0^\circ$ latitude and $110^\circ \text{ E}$ longitude. Billy is standing at point $B$ near Big Baldy Mountain, Idaho, USA, $45^\circ \text{ N}$ latitude and $115^\circ \text{ W}$ longitude. Assuming the Earth is a perfect sphere with center $C$, determine the degree measure of $\angle ACB$.


Suppose  $a$ ,  $b$ ,  $c$ , and  $d$  are non-negative integers such that
    \[(a+b+c+d)(a^2+b^2+c^2+d^2)^2=2023.\]
    Find  $a^3+b^3+c^3+d^3$ .

*Proposed by Connor Gordon*


Let  $V$  be a rectangular prism with integer side lengths. The largest face has area  $240$  and the smallest face has area  $48$ . A third face has area  $x$ , where  $x$  is not equal to  $48$  or  $240$ . What is the sum of all possible values of  $x$ ?


Ana & Bruno decide to play a game with the following rules.:
a) Ana has cards  $1, 3, 5,7,..., 2n-1$ b) Bruno has cards  $2, 4,6, 8,...,2n$ During the first turn and all odd turns afterwards, Bruno chooses one of his cards first and reveals it to Ana, and Ana chooses one of her cards second. Whoever's card is higher gains a point. During the second turn and all even turns afterwards, Ana chooses one of her cards first and reveals it to Bruno, and Bruno chooses one of his cards second. Similarly, whoever's card is higher gains a point. During each turn, neither player can use a card they have already used on a previous turn. The game ends when all cards have been used after  $n$  turns. Determine the highest number of points Ana can earn, and how she manages to do this.

Solution: \boxed{\left\lfloor \frac{n}{2} \right\rfloor}


Given the numbers $a_1,a_2,a_3,a_4,a_5,a_6 \in \{-1,0,1 \}$, find the total count of values of the expression $a_1\cdot 5^1+a_2\cdot 5^2+a_3\cdot 5^3+a_4\cdot 5^4+a_5\cdot 5^5+a_6\cdot 5^6$ that are negative.


Suppose  $ P(x) \equal{} a_nx^n\plus{}\cdots\plus{}a_1x\plus{}a_0$  be a real polynomial of degree  $ n > 2$  with  $ a_n \equal{} 1$ ,  $ a_{n\minus{}1} \equal{} \minus{}n$ ,  $ a_{n\minus{}2} \equal{}\frac{n^2 \minus{} n}{2}$  such that all the roots of  $ P$  are real. Determine the coefficients  $ a_i$ .

Solution: \boxed{a_i = (-1)^{n-i} \binom{n}{i}}


Let  $f:\mathbb{N} \rightarrow \mathbb{N},$   $f(n)=n^2-69n+2250$  be a function. Find the prime number  $p$ , for which the sum of the digits of the number  $f(p^2+32)$  is as small as possible.


Let  $u$  and  $v$  be integers satisfying  $0<v<u.$  Let  $A=(u,v),$  let  $B$  be the reflection of  $A$  across the line  $y=x,$  let  $C$  be the reflection of  $B$  across the y-axis, let  $D$  be the reflection of  $C$  across the x-axis, and let  $E$  be the reflection of  $D$  across the y-axis. The area of pentagon  $ABCDE$  is 451. Find  $u+v.$


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


For  $ n \in \mathbb{N}$ , let  $ f(n)\equal{}1^n\plus{}2^{n\minus{}1}\plus{}3^{n\minus{}2}\plus{}...\plus{}n^1$ . Determine the minimum value of:  $ \frac{f(n\plus{}1)}{f(n)}.$


Suppose that $x, y, z$ are three distinct prime numbers such that $x + y + z = 49$. Find the maximum possible value for the product $xyz$.


Given acute triangle  $\triangle ABC$  in plane  $P$ , a point  $Q$  in space is defined such that  $\angle AQB = \angle BQC = \angle CQA = 90^\circ.$   Point  $X$  is the point in plane  $P$  such that  $QX$  is perpendicular to plane  $P$ . Given  $\angle ABC = 40^\circ$  and  $\angle ACB = 75^\circ,$  find  $\angle AXC.$


Let  $n>2$  be an integer. In a country there are  $n$  cities and every two of them are connected by a direct road. Each road is assigned an integer from the set  $\{1, 2,\ldots ,m\}$  (different roads may be assigned the same number). The *priority* of a city is the sum of the numbers assigned to roads which lead to it. Find the smallest  $m$  for which it is possible that all cities have a different priority.

Solution: \boxed{m = 3}


Find the smallest positive real  $\alpha$ , such that   $$ \frac{x+y} {2}\geq \alpha\sqrt{xy}+(1 - \alpha)\sqrt{\frac{x^2+y^2}{2}} $$   for all positive reals  $x, y$ .

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


Let  $S$  be the set of all partitions of  $2000$  (in a sum of positive integers). For every such partition  $p$ , we deﬁne  $f (p)$  to be the sum of the number of summands in  $p$  and the maximal summand in  $p$ . Compute the minimum of  $f (p)$  when  $p \in S .$


$a,b,c,x,y,z$  are positive real numbers and  $bz+cy=a$ ,  $az+cx=b$ ,  $ay+bx=c$ . Find the least value of following function $f(x,y,z)=\frac{x^2}{1+x}+\frac{y^2}{1+y}+\frac{z^2}{1+z}$


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

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


Given  $\triangle ABC$  with  $\angle A = 15^{\circ}$ , let  $M$  be midpoint of  $BC$  and let  $E$  and  $F$  be points on ray $BA$  and  $CA$  respectively such that  $BE = BM = CF$ . Let  $R_1$  be the radius of  $(MEF)$  and  $R_{2}$  be
radius of  $(AEF)$ . If  $\frac{R_1^2}{R_2^2}=a-\sqrt{b+\sqrt{c}}$  where  $a,b,c$  are integers. Find  $a^{b^{c}}$


Real numbers  $x, y$  satisfy the inequality  $x^2 + y^2 \le 2$ . Orove that  $xy + 3 \ge 2x + 2y$

Solution: \boxed{xy + 3 \ge 2x + 2y}


Evaluate   $ \int_0^1 (1 \plus{} x \plus{} x^2 \plus{} \cdots \plus{} x^{n \minus{} 1})\{1 \plus{} 3x \plus{} 5x^2 \plus{} \cdots \plus{} (2n \minus{} 3)x^{n \minus{} 2} \plus{} (2n \minus{} 1)x^{n \minus{} 1}\}\ dx.$


Determine the smallest positive real number  $r$  such that there exist differentiable functions  $f: \mathbb{R} \rightarrow \mathbb{R}$  and  $g: \mathbb{R} \rightarrow \mathbb{R}$  satisfying
(a)  $f(0)>0$ ,
(b)  $g(0)=0$ ,
(c)  $\left|f^{\prime}(x)\right| \leq|g(x)|$  for all  $x$ ,
(d)  $\left|g^{\prime}(x)\right| \leq|f(x)|$  for all  $x$ , and
(e)  $f(r)=0$ .


For every positive integeer  $n>1$ , let  $k(n)$  the largest positive integer  $k$  such that there exists a positive integer  $m$  such that  $n = m^k$ .

Find   $$ lim_{n \rightarrow \infty} \frac{\sum_{j=2}^{j=n+1}{k(j)}}{n} $$


How many primes $p$ are there such that the number of positive divisors of $p^2+23$ is equal to 14?


A circle of radius $ 1$  is surrounded by  $ 4$  circles of radius  $ r$. What is the value of $ r$?

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


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


Let  $ABC$  be a triangle such that midpoints of three altitudes are collinear. If the largest side of the triangle is  $10$, determine the largest possible area of the triangle.


Find the number of distinct integral solutions of $ x^{4} \plus{}2x^{3} \plus{}3x^{2} \minus{}x\plus{}1\equiv 0\, \, \left(mod\, 30\right)$ where $ 0\le x<30$.


Line $ l_2$ intersects line $ l_1$ and line $ l_3$ is parallel to $ l_1$. The three lines are distinct and lie in a plane. Determine the number of points equidistant from all three lines.


What is the probability of having 2 adjacent white balls or 2 adjacent blue balls in a random arrangement of 3 red, 2 white and 2 blue balls?


Convex pentagon  $ABCDE$  has side lengths  $AB=5$ ,  $BC=CD=DE=6$ , and  $EA=7$ . Moreover, the pentagon has an inscribed circle (a circle tangent to each side of the pentagon). Find the area of  $ABCDE$ .


What is the minimum distance between  $(2019, 470)$  and  $(21a - 19b, 19b + 21a)$  for  $a, b \in Z$ ?


There are  $2024$  cities in a country, every two of which are bidirectionally connected by exactly one of three modes of transportation - rail, air, or road. A tourist has arrived in this country and has the entire transportation scheme. He chooses a travel ticket for one of the modes of transportation and the city from which he starts his trip. He wants to visit as many cities as possible, but using only the ticket for the specified type of transportation. What is the largest  $k$  for which the tourist will always be able to visit at least  $k$  cities? During the route, he can return to the cities he has already visited.

*Proposed by Bogdan Rublov*


Find all natural numbers  $ n $  for which there exists two natural numbers  $ a,b $  such that  $$  n=S(a)=S(b)=S(a+b) ,  $$  where  $ S(k) $  denotes the sum of the digits of  $ k $  in base  $ 10, $  for any natural number  $ k. $ *Vasile Zidaru* and *Mircea Lascu*


1. The numbers  $1,2, \ldots$  are placed in a triangle as following:

\[
\begin{matrix}
1 &  &  &
2 & 3 &  &
4 & 5 & 6 &
7 & 8 & 9 & 10
 \ldots
\end{matrix}
\]

What is the sum of the numbers on the  $n$ -th row?

Solution: \boxed{\frac{n(n^2 + 1)}{2}}


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


$ABCDEF$  is a regular hexagon. Let  $R$  be the overlap between  $\vartriangle ACE$  and  $\vartriangle BDF$ . What is the area of  $R$  divided by the area of  $ABCDEF$ ?


Consider the recurrence relation  $$ a_{n+3}=\frac{a_{n+2}a_{n+1}-2}{a_n} $$  with initial condition  $(a_0,a_1,a_2)=(1,2,5)$ . Let  $b_n=a_{2n}$  for nonnegative integral  $n$ . It turns out that  $b_{n+2}+xb_{n+1}+yb_n=0$  for some pair of real numbers  $(x,y)$ . Compute  $(x,y)$ .

Solution: \boxed{(-4, 1)}


Let  $\mathcal{S}$  be the set  $\{1,2,3,\ldots,10\}.$  Let  $n$  be the number of sets of two non-empty disjoint subsets of  $\mathcal{S}.$  (Disjoint sets are defined as sets that have no common elements.) Find the remainder obtained when  $n$  is divided by  $1000.$


In right triangle ABC with legs 5 and 12, arcs of circles are drawn, one with center A and radius 12, the other with center B and radius 5. They intersect the hypotenuse at M and N. Find the length of MN.


Magicman and his helper want to do some magic trick. They have special card desk. Back of all cards is common color and face is one of  $2017$  colors.
Magic trick: magicman go away from scene. Then viewers should put on the table  $n>1$  cards in the row face up. Helper looks at these cards, then he turn all cards face down, except one, without changing order in row. Then magicman returns on the scene, looks at cards, then show on the one  card, that lays face down and names it face color.
What is minimal  $n$  such that magicman and his helper can has strategy to make magic trick successfully?

Solution: \boxed{n = 2018}


What is the smallest positive integer  $n$  such that  $2013^n$  ends in  $001$  (i.e. the rightmost three digits of  $2013^n$  are  $001$ ?


Find the greatest value of  $M$   $\in \mathbb{R}$  such that the following inequality is true  $\forall$    $x, y, z$    $\in \mathbb{R}$  $x^4+y^4+z^4+xyz(x+y+z)\geq M(xy+yz+zx)^2$ .


$\triangle ABC$  has side lengths  $AB=20$ ,  $BC=15$ , and  $CA=7$ . Let the altitudes of  $\triangle ABC$  be  $AD$ ,  $BE$ , and  $CF$ . What is the distance between the orthocenter (intersection of the altitudes) of  $\triangle ABC$  and the incenter of  $\triangle DEF$ ?


4. Suppose that  $\overline{A2021B}$  is a six-digit integer divisible by  $9$ . Find the maximum possible value of  $A \cdot B$ .

5. In an arbitrary triangle, two distinct segments are drawn from each vertex to the opposite side. What is the minimum possible number of intersection points between these segments?

6. Suppose that  $a$  and  $b$  are positive integers such that  $\frac{a}{b-20}$  and  $\frac{b+21}{a}$  are positive integers. Find the maximum possible value of  $a + b$ .


Peter is chasing after Rob.  Rob is running on the line  $y=2x+5$  at a speed of  $2$  units a second, starting at the point  $(0,5)$ .  Peter starts running  $t$  seconds after Rob, running at  $3$  units a second.  Peter also starts at  $(0,5)$  and catches up to Rob at the point  $(17,39)$ .  What is the value of t?


$K, L, M$  and  $N$  are points on sides  $AB, BC, CD$  and  $DA$ , respectively, of the unit square  $ABCD$  such that  $KM$  is parallel to  $BC$  and  $LN$  is parallel to  $AB$ . The perimeter of triangle  $KLB$  is equal to  $1$ . What is the area of triangle  $MND$ ?


Let  $n$  be a positive integer. Positive numbers  $a$ ,  $b$ ,  $c$  satisfy  $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1$ . Find the greatest possible value of   $$ E(a,b,c)=\frac{a^{n}}{a^{2n+1}+b^{2n} \cdot c + b \cdot c^{2n}}+\frac{b^{n}}{b^{2n+1}+c^{2n} \cdot a + c \cdot a^{2n}}+\frac{c^{n}}{c^{2n+1}+a^{2n} \cdot b + a \cdot b^{2n}} $$


Four disks with disjoint interiors are mutually tangent. Three of them are equal in size and the fourth one is smaller. Find the ratio of the radius of the smaller disk to one of the larger disks.

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


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


Let  $M$  be a finite subset of the plane such that for any two different points  $A,B\in M$  there is a point  $C\in M$  such that  $ABC$  is equilateral. What is the maximal number of points in  $M?$


Let Akbar and Birbal together have  $n$  marbles, where  $n > 0$ .
Akbar says to Birbal, “ If I give you some marbles then you will have twice as many marbles as I will have.”
Birbal says to Akbar, “ If I give you some marbles then you will have thrice as many marbles as I will have.”
What is the minimum possible value of  $n$  for which the above statements are true?


In parallelogram  $ABCD$ , the longer side is twice the shorter side. Let  $XYZW$  be the quadrilateral formed by the internal bisectors of the angles of  $ABCD$ . If the area of  $XYZW$  is  $10$ , find the area of  $ABCD$


How many of the numbers $2,6,12,20,\ldots,14520$ are divisible by 120?


Find all  $t\in \mathbb Z$  such that: exists a function  $f:\mathbb Z^+\to \mathbb Z$  such that: $f(1997)=1998$  $\forall x,y\in \mathbb Z^+ , \text{gcd}(x,y)=d : f(xy)=f(x)+f(y)+tf(d):P(x,y)$

Solution: \boxed{t = -1}


The fraction  $\tfrac1{2015}$  has a unique "(restricted) partial fraction decomposition'' of the form \[\dfrac1{2015}=\dfrac a5+\dfrac b{13}+\dfrac c{31},\] where  $a$ ,  $b$ , and  $c$  are integers with  $0\leq a<5$  and  $0\leq b<13$ .  Find  $a+b$ .


Consider all possible integers  $n \ge 0$  such that  $(5 \cdot 3^m) + 4 = n^2$  holds for some corresponding integer  $m \ge 0$ . Find the sum of all such  $n$ .


In triangle  $ABC$ , let  $J$  be the center of the excircle tangent to side  $BC$  at  $A_{1}$  and to the extensions of the sides  $AC$  and  $AB$  at  $B_{1}$  and  $C_{1}$  respectively. Suppose that the lines  $A_{1}B_{1}$  and  $AB$  are perpendicular and intersect at  $D$ . Let  $E$  be the foot of the perpendicular from  $C_{1}$  to line  $DJ$ . Determine the angles  $\angle{BEA_{1}}$  and  $\angle{AEB_{1}}$ .

*Proposed by Dimitris Kontogiannis, Greece*

Solution: \boxed{\angle BEA_1 = \angle AEB_1 = 90^\circ}


$ABCDEF GH$  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  $IF$ ?


Suppose that  $a_0=1$  and that  $a_{n+1}=a_n+e^{-a_n}$  for  $n=0,1,2,\dots.$  Does  $a_n-\log n$  have a finite limit as  $n\to\infty?$  (Here  $\log n=\log_en=\ln n.$ )

Solution: \boxed{\lim_{n \to \infty} (a_n - \log n) = 0}


Given a square with sides of length $\frac{2}{\pi}$, find the perimeter of the region bounded by the semicircular arcs constructed on the sides of the square.


Let  $n \ge 3$  be an integer. Find the minimal value of the real number  $k_n$  such that for all positive numbers  $x_1, x_2, ..., x_n$  with product  $1$ , we have   $$ \frac{1}{\sqrt{1 + k_nx_1}}+\frac{1}{\sqrt{1 + k_nx_2}}+ ... + \frac{1}{\sqrt{1 + k_nx_n}} \le n - 1. $$


In $\triangle ABC$, $AB=AC=28$ and $BC=20$. Points $D$, $E$, and $F$ are on sides $\overline{AB}$, $\overline{BC}$, and $\overline{AC}$, respectively, such that $\overline{DE}$ and $\overline{EF}$ are parallel to $\overline{AC}$ and $\overline{AB}$, respectively. What is the perimeter of parallelogram $ADEF$?


Let  $a_n$  be a sequence defined by some  $a_0$  and the recursion  $a_{n+1} = a_n + 2 \cdot  3^n$  for  $n \ge 0$ .
Determine all rational values of  $a_0$  such that  $a^j_k / a^k_j$  is an integer for all integers  $j$  and  $k$  with  $0 < j < k$ .

Solution: \boxed{a_0 = 1}


Let  $O$  be the center of the circle  $\omega$  circumscribed around the acute-angled triangle  $\vartriangle ABC$ , and  $W$  be the midpoint of the arc  $BC$  of the circle   $\omega$ , which does not contain the point  $A$ , and  $H$  be the point of intersection of the heights of the triangle   $\vartriangle ABC$ . Find the angle  $\angle BAC$ , if  $WO = WH$ .

(O. Clurman)

Solution: \boxed{\angle BAC = 60^\circ}


Given a positive integer  $n,$  let  $s(n)$  denote the sum of the digits of  $n.$  Compute the largest positive integer  $n$  such that  $n = s(n)^2 + 2s(n) - 2.$


Find all pairs of real number  $x$  and  $y$  which simultaneously satisfy the following 2 relations: $x+y+4=\frac{12x+11y}{x^2+y^2}$  $y-x+3=\frac{11x-12y}{x^2+y^2}$

Solution: \boxed{(2, 1)}


Define  $ a \circledast b = a + b-2ab $ . Calculate the value of  $$ A=\left( ...\left(\left(\frac{1}{2014}\circledast \frac{2}{2014}\right)\circledast\frac{3}{2014}\right)...\right)\circledast\frac{2013}{2014} $$


The polynomial  $ x^3\minus{}ax^2\plus{}bx\minus{}2010$  has three positive integer zeros. Determine the smallest possible value of  $ a$.


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  $m+n$ ?


A rectangle can be divided into  $n$  equal squares. The same rectangle can also be divided into  $n+76$  equal squares. Find  $n$ .


If facilities for division are not available, it is sometimes convenient in determining the decimal expansion of  $1/a$ ,  $a>0$ , to use the iteration   $$ x_{k+1}=x_k(2-ax_k), \quad \quad k=0,1,2,\dots , $$   where  $x_0$  is a selected “starting” value. Find the limitations, if any, on the starting values  $x_0$ , in order that the above iteration converges to the desired value  $1/a$ .

Solution: \boxed{0 < x_0 < \frac{2}{a}}


Given that $\sin x \cos x \leq C(\sin^6x+\cos^6x)$ for every real number $x$, determine the least real number $C$.


Find the area of quadrilateral  $ABCD$  if: two opposite angles are right;two sides which form right angle are of equal length and sum of lengths of other two sides is  $10$


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})}}$  -  $\sum_{k=1}^{n-1} (1-\frac {1} {n})	\dots(1-\frac{k}{n})$


For a natural number  $k>1$ , define  $S_k$  to be the set of all triplets  $(n,a,b)$  of natural numbers, with  $n$  odd and  $\gcd (a,b)=1$ , such that  $a+b=k$  and  $n$  divides  $a^n+b^n$ . Find all values of  $k$  for which  $S_k$  is finite.

Solution: \boxed{k = 2^m}


Consider decompositions of an  $8\times 8$  chessboard into  $p$  non-overlapping rectangles subject to the following conditions:
(i) Each rectangle has as many white squares as black squares.
(ii) If  $a_i$  is the number of white squares in the  $i$ -th rectangle, then  $a_1<a_2<\ldots <a_p$ .
Find the maximum value of  $p$  for which such a decomposition is possible. For this value of  $p$ , determine all possible sequences  $a_1,a_2,\ldots ,a_p$ .

Solution: \boxed{p = 7}


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

A. Gribalko


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

Solution: \boxed{x = 3}


If  $a!+\left(a+2\right)!$  divides  $\left(a+4\right)!$  for some nonnegative integer  $a$ , what are all possible values of  $a$ ?

*2019 CCA Math Bonanza Individual Round #8*

Solution: \boxed{0, 3}


The quadratic polynomial  $f(x)$  has the expansion  $2x^2 - 3x + r$ . What is the largest real value of  $r$  for which the ranges of the functions  $f(x)$  and  $f(f(x))$  are the same set?


Given that the inequality $1+\sqrt{n^2-9n+20} > \sqrt{n^2-7n+12}$ holds true, determine the number of positive integer $n$ satisfying this inequality.


Solution: \boxed{x_1 = x_2 = x_3 = x_4 = x_5}


$N\geq9$  distinct real numbers are written on a blackboard. All these numbers are nonnegative, and all are less than  $1$ . It happens that for very  $8$  distinct numbers on the board, the board contains the ninth number distinct from eight such that the sum of all these nine numbers is integer. Find all values  $N$  for which this is possible. *(F. Nilov)*

Solution: \boxed{N = 9}


Determine all functions  $f: \mathbb{R} \rightarrow \mathbb{R}$  which satisfy conditions: $f(x) + f(y) \ge xy$  for all real  $x, y$  and
for each real  $x$  exists real  $y$ , such that  $f(x) + f(y) = xy$ .

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


Find all positive integers  $a$ ,  $b$ ,  $n$  and prime numbers  $p$  that satisfy

\[	a^{2013} + b^{2013} = p^n\text{.}\]

*Proposed by Matija Bucić.*

Solution: \boxed{(a, b, n, p) = (2^k, 2^k, 2013k + 1, 2)}


Let  $x$  and  $y$  be two real numbers such that  $2 \sin x \sin y + 3 \cos y + 6 \cos x \sin y = 7$ . Find  $\tan^2 x + 2 \tan^2 y$ .


Define a function  $f:\mathbb{N}\rightarrow\mathbb{N}$ , \[f(1)=p+1,\] \[f(n+1)=f(1)\cdot f(2)\cdots f(n)+p,\] where  $p$  is a prime number. Find all  $p$  such that there exists a natural number  $k$  such that  $f(k)$  is a perfect square.

Solution: \boxed{p = 3}


Alan, Jason, and Shervin are playing a game with MafsCounts questions. They each start with  $2$  tokens. In each round, they are given the same MafsCounts question. The first person to solve the MafsCounts question wins the round and steals one token from each of the other players in the game. They all have the same probability of winning any given round. If a player runs out of tokens, they are removed from the game. The last player remaining wins the game.

If Alan wins the first round but does not win the second round, what is the probability that he wins the game?

*2020 CCA Math Bonanza Individual Round #4*


Let  $f(x) = x^2 + 6x + c$  for all real number s $x$ , where  $c$  is some real number.  For what values of  $c$  does  $f(f(x))$  have exactly  $3$  distinct real roots?

Solution: \boxed{c = \frac{11 - \sqrt{13}}{2}}


Find the minimum value of  $c$  such that for any positive integer  $n\ge 4$  and any set  $A\subseteq \{1,2,\cdots,n\}$ , if  $|A| >cn$ , there exists a function  $f:A\to\{1,-1\}$  satisfying  $$ \left| \sum_{a\in A}a\cdot f(a)\right| \le 1. $$


On Binary Island, residents communicate using special paper. Each piece of paper is a  $1 \times n$  row of initially uncolored squares. To send a message, each square on the paper must either be colored either red or green. Unfortunately the paper on the island has become damaged, and each sheet of paper has  $10$  random consecutive squares each of which is randomly colored red or green.

Malmer and Weven would like to develop a scheme that allows them to send messages of length  $2016$  between one another. They would like to be able to send any message of length  $2016$ , and they want their scheme to work with perfect accuracy. What is the smallest value of  $n$  for which they can develop such a strategy?

*Note that when sending a message, one can see which 10 squares are colored and what colors they are. One also knows on which square the message begins, and on which square the message ends.*


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

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


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.


A $2.00 \text{L}$  balloon at  $20.0^{\circ} \text{C}$   and  $745 \text{mmHg}$  floats to an altitude where the temperature is  $10.0^{\circ} \text{C}$  and the air pressure is  $700 \text{mmHg}$ . Calculate the new volume of the balloon.

Solution: \boxed{2.06 \text{ L}}


Find all differentiable functions  $f\colon [0,\infty)\to [0,\infty)$  for which  $f(0)=0$  and  $f^{\prime}(x^2)=f(x)$  for any  $x\in [0,\infty)$  .

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


The positive integer  $ n$  has the following property:
if the three last digits of  $n$  are removed, the number  $\sqrt[3]{n}$  remains.
Find  $n$ .


We are given a convex quadrilateral  $ABCD$  in the plane.
(*i*) If there exists a point  $P$  in the plane such that the areas of  $\triangle ABP, \triangle BCP, \triangle CDP, \triangle DAP$  are equal, what condition must be satisfied by the quadrilateral  $ABCD$ ?
(*ii*) Find (with proof) the maximum possible number of such point  $P$  which satisfies the condition in (*i*).


Source: 1976 Euclid Part B Problem 2
-----
Given that  $x$ ,  $y$ , and  $2$  are in geometric progression, and that  $x^{-1}$ ,  $y^{-1}$ , and  $9x^{-2}$  are in are in arithmetic progression, then find the numerical value of  $xy$ .


$ P(x)$  is a quadratic trinomial. What maximum number of terms equal to the sum of the two preceding terms can occur in the sequence  $ P(1)$ ,  $ P(2)$ ,  $ P(3)$ ,  $ \dots?$

*Proposed by A. Golovanov*


Find all real numbers  $a$  for which there exists a non-constant function  $f :\Bbb R \to \Bbb R$  satisfying the following two equations for all  $x\in \Bbb R:$ i)  $f(ax) = a^2f(x)$  and
ii)  $f(f(x)) = a f(x).$

Solution: \boxed{0, 1}


Find all triples  $(a, b, c)$  of positive integers for which   $$ \begin{cases} a + bc=2010  b + ca = 250\end{cases} $$

Solution: \boxed{(3, 223, 9)}


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


Find the maximum value of real number  $A$  such that   $$ 3x^2 + y^2 + 1 \geq A(x^2 + xy + x) $$   for all positive integers  $x, y.$


Compute the determinant of the  $n\times n$  matrix  $A=(a_{ij})_{ij}$ ,  $$ a_{ij}=\begin{cases}
(-1)^{|i-j|} & \text{if}\, i\ne j,
2 & \text{if}\, i= j.
\end{cases} $$

Solution: \boxed{n + 1}


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

Solution: \boxed{\lambda = 1}


Given real numbers $a$ and $b$ are chosen with $1<a<b$ such that no triangle with positive area has side lengths $1,a,$ and $b$ or $\tfrac{1}{b}, \tfrac{1}{a},$ and $1$, find the smallest possible value of $b$.

Solution: \boxed{\frac{3 + \sqrt{5}}{2}}


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


Given $a,b,c$ are distinct real roots of $x^3-3x+1=0$, calculate the value of $a^8+b^8+c^8$.


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


Assuming that the roots of  $x^3 +px^2 +qx +r=0$  are all real and positive, find the relation between  $p,q,r$  which is a necessary and sufficient condition that the roots are the cosines of the angles of a triangle.

Solution: \boxed{p^2 - 2q - 2r - 1 = 0}


Let  $\alpha$  be an arbitrary positive real number. Determine for this number  $\alpha$  the greatest real number  $C$  such that the inequality  $$ \left(1+\frac{\alpha}{x^2}\right)\left(1+\frac{\alpha}{y^2}\right)\left(1+\frac{\alpha}{z^2}\right)\geq C\left(\frac{x}{z}+\frac{z}{x}+2\right) $$  is valid for all positive real numbers  $x, y$  and  $z$  satisfying  $xy + yz + zx =\alpha.$   When does equality occur?
*(Proposed by Walther Janous)*


The points  $P$  and  $Q$  are the midpoints of the edges  $AE$  and  $CG$  on the cube  $ABCD.EFGH$  respectively. If the length of the cube edges is  $1$  unit, determine the area of the quadrilateral  $DPFQ$  .


The cells of a  $2021\times 2021$  table are filled with numbers using the following rule. The bottom left cell, which we label with coordinate  $(1, 1)$ , contains the number  $0$ . For every other cell  $C$ , we consider a route from  $(1, 1)$  to  $C$ , where at each step we can only go one cell to the right or one cell up (not diagonally). If we take the number of steps in the route and add the numbers from the cells along the route, we obtain the number in cell  $C$ . For example, the cell with coordinate  $(2, 1)$  contains  $1 = 1 + 0$ , the cell with coordinate  $(3, 1)$  contains  $3 = 2 + 0 + 1$ , and the cell with coordinate  $(3, 2)$  contains  $7 = 3 + 0 + 1 + 3$ . What is the last digit of the number in the cell  $(2021, 2021)$ ?


Let  $ABCD$  be a quadrilateral with  $AD = 20$  and  $BC = 13$ .  The area of  $\triangle ABC$  is  $338$  and the area of  $\triangle DBC$  is  $212$ .  Compute the smallest possible perimeter of  $ABCD$ .

*Proposed by Evan Chen*