Let $\mathbb{N}_0$ and $\mathbb{Z}$ be the set of all non-negative integers and the set of all integers, respectively. Let $f:\mathbb{N}_0\rightarrow\mathbb{Z}$ be a function defined as \[f(n)=-f\left(\left\lfloor\frac{n}{3}\right\rfloor \right)-3\left\{\frac{n}{3}\right\} \] where $\lfloor x \rfloor$ is the greatest integer smaller than or equal to $x$ and $\{ x\}=x-\lfloor x \rfloor$. Find the smallest integer $n$ such that $f(n)=2010$.

Let $f,g$ be polynomials with complex coefficients such that $\gcd(\deg f,\deg g)=1$. Suppose that there exist polynomials $P(x,y)$ and $Q(x,y)$ with complex coefficients such that $f(x)+g(y)=P(x,y)Q(x,y)$. Show that one of $P$ and $Q$ must be constant. [i]Victor Wang.[/i]

Let $S$ be the set of all lattice points $(x, y)$ in the plane satisfying $|x|+|y|\le 10$. Let $P_1,P_2,\ldots,P_{2013}$ be a sequence of 2013 (not necessarily distinct) points such that for every point $Q$ in $S$, there exists at least one index $i$ such that $1\le i\le 2013$ and $P_i = Q$. Suppose that the minimum possible value of $|P_1P_2|+|P_2P_3|+\cdots+|P_{2012}P_{2013}|$ can be expressed in the form $a+b\sqrt{c}$, where $a,b,c$ are positive integers and $c$ is not divisible by the square of any prime. Find $a+b+c$. (A [i]lattice point[/i] is a point with all integer coordinates.) [hide="Clarifications"] [list] [*] $k = 2013$, i.e. the problem should read, ``... there exists at least one index $i$ such that $1\le i\le 2013$ ...''. An earlier version of the test read $1 \le i \le k$.[/list][/hide] [i]Anderson Wang[/i]

What is the largest quotient that can be formed using two numbers chosen from the set $\{ -24, -3, -2, 1, 2, 8 \}$? $\text{(A)}\ -24 \qquad \text{(B)}\ -3 \qquad \text{(C)}\ 8 \qquad \text{(D)}\ 12 \qquad \text{(E)}\ 24$

Find all positive integers $n$ and $b$ with $0 < b < 10$ such that if $a_n$ is the positive integer with $n$ digits, all of them $1$, then $a_{2n} - b a_n$ is a square.

In the plane, we have any polygon that does not intersect itself and is closed. Given a point that is not on the edge of the polygon. How can we determine whether it is inside or outside the polygon? (the polygon has a finite number of sides) [hide=original wording]En el plano se tiene un poligono cualquiera que no se corta a si mismo y que es cerrado. Dado un punto que no esta sobre el borde del poligono, Como determinara se esta dentro o fuera del poligono? (el poligono tiene un numero nito de lados)[/hide]

In the plane, there are $n$ points ($n\ge 4$) where no 3 of them are collinear. Let $A(n)$ be the number of parallelograms whose vertices are those points with area $1$. Prove the following inequality: $A(n)\leq \frac{n^2-3n}{4}$ for all $n\ge 4$

The $4$ code words $$\square * \otimes \,\,\,\, \oplus \rhd \bullet \,\,\,\, * \square \bullet \,\,\,\, \otimes \oslash \oplus$$ they are in some order $$AMO \,\,\,\, SUR \,\,\,\, REO \,\,\,\, MAS$$ Decrypt $$\otimes \oslash \square * \oplus \rhd \square \bullet \otimes $$

Hour part of a defective digital watch displays only the numbers from $1$ to $12$. After one minute from $ n: 59$, although it must display $ \left(n \plus{} 1\right): 00$, it displays $ 2n: 00$ (Think in $ mod\, 12$). For example, after $ 7: 59$, it displays $ 2: 00$ instead of $ 8: 00$. If we set the watch to an arbitrary time, what is the probability that hour part displays $4$ after exactly one day? $\textbf{(A)}\ \frac {1}{12} \qquad\textbf{(B)}\ \frac {1}{4} \qquad\textbf{(C)}\ \frac {1}{3} \qquad\textbf{(D)}\ \frac {1}{2} \qquad\textbf{(E)}\ \text{None}$

If $a$ and $b$ are nonzero real numbers such that $\left| a \right| \ne \left| b \right|$, compute the value of the expression \[ \left( \frac{b^2}{a^2} + \frac{a^2}{b^2} - 2 \right) \times \left( \frac{a + b}{b - a} + \frac{b - a}{a + b} \right) \times \left( \frac{\frac{1}{a^2} + \frac{1}{b^2}}{\frac{1}{b^2} - \frac{1}{a^2}} - \frac{\frac{1}{b^2} - \frac{1}{a^2}}{\frac{1}{a^2} + \frac{1}{b^2}} \right). \]

Let $I$ be the incenter of a tetrahedron $ABCD$, and $J$ be the center of the exsphere touching the face $BCD$ containing three remaining faces (outside these faces). The segment $IJ$ meets the circumsphere of the tetrahedron at point $K$. Which of two segments $IJ$ and $JK$ is longer?

Prove that the number of orientations of a connected $3$-regular graph on $2n$ vertices where the number of vertices with indegree $0$ and outdegree $0$ are equal, is exactly $2^{n+1}$ $ {2n} \choose {n}$.

Find all two-variable polynomials $p(x,y)$ such that for each $a,b,c\in\mathbb R$: \[p(ab,c^2+1)+p(bc,a^2+1)+p(ca,b^2+1)=0\]

We say that we extend a finite sequence of positive integers $(a_1,\dotsc,a_n)$ if we replace it by \[(1,2,\dotsc,a_1-1,a_1,1,2,\dotsc,a_2-1,a_2,1,2,\dotsc,a_3-1,a_3,\dotsc,1,2,\dotsc,a_n-1,a_n)\] i.e., each element $k$ of the original sequence is replaced by $1,2,\dotsc,k$. Géza takes the sequence $(1,2,\dotsc,9)$ and he extends it $2017$ times. Then he chooses randomly one element of the resulting sequence. What is the probability that the chosen element is $1$?

For this exercise, $ \{\} $ denotes the fractional part. [b]a)[/b] Let be a natural number $ n. $ Compare $ \left\{ \sqrt{n+1} -\sqrt{n} \right\} $ with $ \left\{ \sqrt{n} -\sqrt{n-1} \right\} . $ [b]b)[/b] Show that there are two distinct natural numbers $ a,b, $ such that $ \left\{ \sqrt{a} -\sqrt{b} \right\} =\left\{ \sqrt{b} -\sqrt{a} \right\} . $ [i]Traian Preda[/i]

Given a $6\times 6$ square consisting of unit squares, denote its rows and columns from $1$ to $6$. Figure [i]p-horse[/i] can move from square $(x; y)$ to $(x’; y’)$ if and only if both $x + x’$ and $y + y’$ are primes. At the start the [i]p-horse[/i] is located in one of the unit squares. $a)$ Can the [i]p-horse[/i] visit every unit square exactly once? $b$) Can the [i]p-horse[/i] visit every unit square exactly once and with the last move return to the initial starting position?

Let $A_1B_1C_1$ be a triangle with $A_1B_1 = 16, B_1C_1 = 14,$ and $C_1A_1 = 10$. Given a positive integer $i$ and a triangle $A_iB_iC_i$ with circumcenter $O_i$, define triangle $A_{i+1}B_{i+1}C_{i+1}$ in the following way: (a) $A_{i+1}$ is on side $B_iC_i$ such that $C_iA_{i+1}=2B_iA_{i+1}$. (b) $B_{i+1}\neq C_i$ is the intersection of line $A_iC_i$ with the circumcircle of $O_iA_{i+1}C_i$. (c) $C_{i+1}\neq B_i$ is the intersection of line $A_iB_i$ with the circumcircle of $O_iA_{i+1}B_i$. Find \[ \left(\sum_{i = 1}^\infty [A_iB_iC_i] \right)^2. \] Note: $[K]$ denotes the area of $K$. [i]Proposed by Yang Liu[/i]

Let $P$ be an interior point of the triangle $ABC$ which is not on the median belonging to $BC$ and satisfying $\angle CAP = \angle BCP. \: BP \cap CA = \{B'\} \: , \: CP \cap AB = \{C'\}$ and $Q$ is the second point of intersection of $AP$ and the circumcircle of $ABC. \: B'Q$ intersects $CC'$ at $R$ and $B'Q$ intersects the line through $P$ parallel to $AC$ at $S.$ Let $T$ be the point of intersection of lines $B'C'$ and $QB$ and $T$ be on the other side of $AB$ with respect to $C.$ Prove that \[\angle BAT = \angle BB'Q \: \Longleftrightarrow \: |SQ|=|RB'| \]

Find all pairs $(a,b)$ of positive integers such that $a!+b$ and $b!+a$ are both powers of $5$. [i]Nikola Velov, North Macedonia[/i]

Define a function $f$ on the positive integers as follows: $f(n) = m$, where $m$ is the least positive integer such that $n$ is a factor of $m^2$. Find the smallest integer $M$ such that $\sqrt{M}$ is both a product of prime numbers, of which there are at least $3$, and a factor of $$\sum_{ d|M} f(d),$$ the sum of $f(d)$ for all positive integers $d$ that divide $M$.

Let be a $ 3\times 3 $ complex matrix such that $ A^3=I $ and for which exist four real numbers $ a,b,c,d $ with $ a,c\neq 1 $ such that $ \det \left( A^2+aA+bI \right) =\det \left( A^2+cA+dI \right) =0. $ Show that $ a+b=c+d. $ [i]C. Merticaru[/i]

$ABCD$ is a quadrilateral with a circumcircle centre $O$ and an inscribed circle centre $I$. The diagonals intersect at $S$. Show that if two of $O,I,S$ coincide, then it must be a square.

For each positive integer $n,$ let $\sigma(n)$ denote the sum of the positive integer divisors of $n.$ How many positive integers $n \leq 2021$ satisfy $$\sigma(3n) \geq \sigma(n)+\sigma(2n)?$$ [i]Proposed by Kyle Lee[/i]

Let $O$, $I$ be the circumcenter and the incenter of an acute-angled scalene triangle $ABC$; $D$, $E$, $F$ be the touching points of its excircle with the side $BC$ and the extensions of $AC$, $AB$ respectively. Prove that if the orthocenter of the triangle $DEF$ lies on the circumcircle of $ABC$, then it is symmetric to the midpoint of the arc $BC$ with respect to $OI$. Proposed by: P.Puchkov,E.Utkin

Let $n\geq 2$ be an integer and let $a_1, a_2, \ldots, a_n$ be positive real numbers with sum $1$. Prove that $$\sum_{k=1}^n \frac{a_k}{1-a_k}(a_1+a_2+\cdots+a_{k-1})^2 < \frac{1}{3}.$$