International Competitions 2023

Determine all composite integers $n > 1$ that satisfy the following property: if $d_1, d_2, \ldots, d_k$ are all the positive divisors of $n$ with $1 = d_1 < d_2 < \cdots < d_k = n$, then $d_i$ divides $d_{i+1} + d_{i+2}$ for every $1 \leqslant i \leqslant k-2$.

Let $ABC$ be an acute-angled triangle with $AB < AC$. Let $\Omega$ be the circumcircle of $ABC$. Let $S$ be the midpoint of the arc $CB$ of $\Omega$ containing $A$. The perpendicular from $A$ to $BC$ meets $BS$ at $D$ and meets $\Omega$ again at $E \neq A$. The line through $D$ parallel to $BC$ meets line $BE$ at $L$. Denote the circumcircle of triangle $BDL$ by $\omega$. Let $\omega$ meet $\Omega$ again at $P \neq B$. Prove that the line tangent to $\omega$ at $P$ meets line $BS$ on the internal angle bisector of $\measuredangle BAC$.

For each integer $k \geqslant 2$, determine all infinite sequences of positive integers $a_1, a_2, \ldots$ for which there exists a polynomial $P$ of the form $P(x) = x^k + c_{k-1}x^{k-1} + \cdots + c_1x + c_0$, where $c_0, c_1, \ldots, c_{k-1}$ are non-negative integers, such that

$$P(a_n) = a_{n+1}a_{n+2}\cdots a_{n+k}$$

for every integer $n \geqslant 1$.

Let $x_1, x_2, \ldots, x_{2023}$ be pairwise different positive real numbers such that

$$a_n = \sqrt{(x_1 + x_2 + \cdots + x_n)\left(\frac{1}{x_1} + \frac{1}{x_2} + \cdots + \frac{1}{x_n}\right)}$$

is an integer for every $n = 1, 2, \ldots, 2023$. Prove that $a_{2023} \geqslant 3034$.

Let $n$ be a positive integer. A Japanese triangle consists of $1 + 2 + \cdots + n$ circles arranged in an equilateral triangular shape such that for each $i = 1, 2, \ldots, n$, the $i^{\text{th}}$ row contains exactly $i$ circles, exactly one of which is coloured red. A ninja path in a Japanese triangle is a sequence of $n$ circles obtained by starting in the top row, then repeatedly going from a circle to one of the two circles immediately below it and finishing in the bottom row. Here is an example of a Japanese triangle with $n = 6$, along with a ninja path in that triangle containing two red circles.

In terms of $n$, find the greatest $k$ such that in each Japanese triangle there is a ninja path containing at least $k$ red circles.

Let $ABC$ be an equilateral triangle. Let $A_1, B_1, C_1$ be interior points of $ABC$ such that $BA_1 = A_1C$, $CB_1 = B_1A$, $AC_1 = C_1B$, and

$$\angle BA_1C + \angle CB_1A + \angle AC_1B = 480°.$$

Let $BC_1$ and $CB_1$ meet at $A_2$, let $CA_1$ and $AC_1$ meet at $B_2$, and let $AB_1$ and $BA_1$ meet at $C_2$. Prove that if triangle $A_1B_1C_1$ is scalene, then the three circumcircles of triangles $AA_1A_2$, $BB_1B_2$ and $CC_1C_2$ all pass through two common points.

(Note: a scalene triangle is one where no two sides have equal length.)

Let $\mathbb{R}$ denote the set of all real numbers. For each pair $(\alpha, \beta)$ of nonnegative real numbers subject to $\alpha + \beta \geq 2$, determine all functions $f\colon \mathbb{R} \to \mathbb{R}$ satisfying

$$f(x)f(y) \leq f(xy) + \alpha x + \beta y$$

for all real numbers $x$ and $y$.

Find all integers $n \geq 3$ for which it is possible to draw $n$ chords of one circle such that their $2n$ endpoints are pairwise distinct and each chord intersects precisely $k$ other chords for:

(a) $k = n - 2$,

(b) $k = n - 3$.

Remark. A chord of a circle is a line segment whose both endpoints lie on the circle.

Let $ABC$ be a triangle with incenter $I$. The incircle $\omega$ of $ABC$ is tangent to the line $BC$ at point $D$. Denote by $E$ and $F$ the points satisfying $AI \parallel BE \parallel CF$ and $\angle BEI = \angle CFI = 90°$. Lines $DE$ and $DF$ intersect $\omega$ again at points $E'$ and $F'$, respectively. Prove that $E'F' \perp AI$.

Let $n$ and $m$ be positive integers. We call a set $S$ of positive integers $(n, m)$-good if it satisfies the following three conditions:

(i) We have $m \in S$.

(ii) For all $a \in S$, all of the positive divisors of $a$ are elements of $S$ too.

(iii) For all mutually different numbers $a, b \in S$, we have $a^n + b^n \in S$.

Determine all pairs $(n, m)$ such that the set of all positive integers is the only $(n, m)$-good set.

Let $\mathbb{Z}$ denote the set of all integers and $\mathbb{Z}_{>0}$ denote the set of all positive integers.

(a) A function $f\colon \mathbb{Z}\to \mathbb{Z}$ is called $\mathbb{Z}$-good if it satisfies $f(a^{2} + b) = f(b^{2} + a)$ for all $a,b\in \mathbb{Z}$. Determine the largest possible number of distinct values that can occur among $f(1),f(2),\ldots ,f(2023)$, where $f$ is a $\mathbb{Z}$-good function.

(b) A function $f\colon \mathbb{Z}_{>0}\to \mathbb{Z}_{>0}$ is called $\mathbb{Z}_{>0}$-good if it satisfies $f(a^{2} + b) = f(b^{2} + a)$ for all $a,b\in \mathbb{Z}_{>0}$. Determine the largest possible number of distinct values that can occur among $f(1),f(2),\ldots ,f(2023)$, where $f$ is a $\mathbb{Z}_{>0}$-good function.

Let $a, b, c$ and $d$ be positive real numbers with $abcd = 1$. Prove that

$$\frac {a b + 1}{a + 1} + \frac {b c + 1}{b + 1} + \frac {c d + 1}{c + 1} + \frac {d a + 1}{d + 1} \geq 4,$$

and determine all quadruples $(a,b,c,d)$ for which equality holds.

Find the smallest integer $b$ with the following property: For each way of colouring exactly $b$ squares of an $8 \times 8$ chessboard green, one can place 7 bishops on 7 green squares so that no two bishops attack each other.

Remark. Two bishops attack each other if they are on the same diagonal.

Let $c \geq 4$ be an even integer. In some football league, each team has a home uniform and an away uniform. Every home uniform is coloured in two different colours, and every away uniform is coloured in one colour. A team's away uniform cannot be coloured in one of the colours from the home uniform. There are at most $c$ distinct colours on all of the uniforms. If two teams have the same two colours on their home uniforms, then they have different colours on their away uniforms.

We say a pair of uniforms is clashing if some colour appears on both of them. Suppose that for every team $X$ in the league, there is no team $Y$ in the league such that the home uniform of $X$ is clashing with both uniforms of $Y$. Determine the maximum possible number of teams in the league.

We are given a convex quadrilateral $ABCD$ whose angles are not right. Assume there are points $P, Q, R, S$ on its sides $AB, BC, CD, DA$, respectively, such that $PS \parallel BD$, $SQ \perp BC$, $PR \perp CD$. Furthermore, assume that the lines $PR, SQ$, and $AC$ are concurrent. Prove that the points $P, Q, R, S$ are concyclic.

Let $ABC$ be an acute triangle with $AB < AC$. Let $J$ be the center of the $A$-excircle of $ABC$. Let $D$ be the projection of $J$ on line $BC$. The internal bisectors of angles $BDJ$ and $JDC$ intersect lines $BJ$ and $JC$ at $X$ and $Y$, respectively. Segments $XY$ and $JD$ intersect at $P$. Let $Q$ be the projection of $A$ on line $BC$. Prove that the internal angle bisector of $\measuredangle QAP$ is perpendicular to line $XY$.

Remark. The $A$-excircle of the triangle $ABC$ is the circle outside the triangle which is tangent to the lines $AB$, $AC$, and the line segment $BC$.

Find all positive integers $n$ for which there exist positive integers $a > b$ satisfying

$$n = \frac{4ab}{a - b}.$$

Let $A$ and $B$ be positive integers. Consider a sequence of positive integers $(x_{n})_{n\geq 1}$ such that

$$x_{n+1} = A \cdot \gcd(x_{n}, x_{n-1}) + B \quad \text{for every } n \geq 2.$$

Prove that the sequence attains only finitely many different values.

Remark. We denote by $\gcd(a, b)$ the greatest common divisor of positive integers $a$ and $b$.