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Stage 10 · Mixed

10 problems · Hardest shortlist tier · mathsolympiadprep.com

The answer key prints on its own page at the end.

Given an integer $n > 1$ and an integer $a$ that is coprime with $n$ . There is a country consisting of $n$ islands $D_1, D_2, \dots, D_n$ . For any two different islands $D_i$ and $D_j$ , there is a one-way ferry from $D_i$ to $D_j$ if and only if $ij \equiv ia \pmod{n}$ . A tourist hopes to visit as many islands as possible. He can first fly to any island he chooses to start the tour, and afterwards can only use the one-way ferry to tour freely between islands in this country. Find the maximum possible number of different islands that the tourist can visit.

Number theory Solution and answer checking →
Let $ABC$ be a triangle and $N$ be a point that differs from $A$ , $B$ and $C$ . Let $A_b$ be the reflection of $A$ through $NB$ , and $B_a$ be the reflection of $B$ through $NA$ . We define $B_c$ , $C_b$ , $A_c$ and $C_a$ similarly. Let $m_a$ be the line passing through $N$ and perpendicular to $B_cC_b$ . Define $m_b$ , $m_c$ similarly.
a) Assume that $N$ is the orthocenter of triangle $ABC$ , show that the respective reflections of the lines $m_a$ , $m_b$ and $m_c$ through each bisector of angles $\angle BNC$ , $\angle CNA$ and $\angle ANB$ are coincident.
b) Assume that $N$ is the nine-point center of triangle $ABC$ , show that the respective reflections of the lines $m_a$ , $m_b$ and $m_c$ through the lines $BC$ , $CA$ and $AB$ concur.

Geometry Solution and answer checking →
Determine all functions $f: \mathbb{Q} \longrightarrow \mathbb{Z}$ satisfying
$f\left(\frac{f(x)+a}{b}\right)=f\left(\frac{x+a}{b}\right)$
for all $x \in \mathbb{Q}$ , $a \in \mathbb{Z}$ , and $b \in \mathbb{Z}_{>0}$ . (Here, $\mathbb{Z}_{>0}$ denotes the set of positive integers.)

Algebra Solution and answer checking →
Let $A_{0} = (a_{1}, \ldots, a_{n})$ be a finite sequence of real numbers. For each $k \geq 0$ , from the sequence $A_{k} = (x_{1}, \ldots, x_{n})$ we construct a new sequence $A_{k+1}$ in the following way.
1. We choose a partition $\{1, \ldots, n\} = I \cup J$ , where $I$ and $J$ are two disjoint sets, such that the expression
$\left|\sum_{i \in I} x_{i} - \sum_{j \in J} x_{j}\right|$
attains the smallest possible value. (We allow the sets $I$ or $J$ to be empty; in this case the corresponding sum is $0$ .) If there are several such partitions, one is chosen arbitrarily.
2. We set $A_{k+1} = (y_{1}, \ldots, y_{n})$ , where $y_{i} = x_{i} + 1$ if $i \in I$ , and $y_{i} = x_{i} - 1$ if $i \in J$ .
Prove that for some $k$ , the sequence $A_{k}$ contains an element $x$ such that $|x| \geq n / 2$ .

Combinatorics Solution and answer checking →
Let $n$ be a positive integer. We say that a polynomial $P$ with integer coefficients is $n$ -good if there exists a polynomial $Q$ of degree 2 with integer coefficients such that $Q(k)(P(k)+Q(k))$ is never divisible by $n$ for any integer $k$ .
Determine all integers $n$ such that every polynomial with integer coefficients is an $n$ -good polynomial.

Number theory Solution and answer checking →
Let $\mathcal{V}$ be a finite set of points in the plane. We say that $\mathcal{V}$ is balanced if for any two distinct points $A, B \in \mathcal{V}$ , there exists a point $C \in \mathcal{V}$ such that $AC = BC$ . We say that $\mathcal{V}$ is center-free if for any distinct points $A, B, C \in \mathcal{V}$ , there does not exist a point $P \in \mathcal{V}$ such that $PA = PB = PC$ .
a. Show that for all $n \geqslant 3$ , there exists a balanced set consisting of $n$ points.
b. For which $n \geqslant 3$ does there exist a balanced, center-free set consisting of $n$ points?

Geometry Solution and answer checking →
Find all positive integers $n \ge 2$ for which there exist $n$ real numbers
$a_1 < a_2 < \dots < a_n$
and a real number $r > 0$ such that the $\frac{n(n-1)}{2}$ differences $a_j - a_i$ for $1 \le i < j \le n$ are equal, in some order, to the numbers
$r^1, r^2, \dots, r^{\frac{n(n-1)}{2}}.$

Algebra Solution and answer checking →
Suppose there are 101 persons sitting around a round table in an arbitrary order. The $k$ th person possesses $k$ pieces of cards, $k = 1, \dots, 101$ . We call it a transition if one transits one of his cards to one of his adjacent persons. Find the minimum positive number $k$ , such that whatever the order of the seating, there is a way of no more than $k$ transitions so that each person possesses 51 cards.

Combinatorics Solution and answer checking →
Given a positive integer $n$ , let $D$ be the set of positive divisors of $n$ , and let $f: D \to \mathbb{Z}$ be a function. Prove that the following are equivalent:
(A) for any positive divisor $m$ of $n$ ,
$n \mid \sum_{d|m} f(d) \binom{n/d}{m/d};$
(B) for any positive divisor $k$ of $n$ ,
$k \mid \sum_{d|k} f(d).$

Number theory Solution and answer checking →
Let $A B C D E$ be a convex pentagon with $C D = D E$ and $\angle E D C \neq 2 \cdot \angle A D B$ . Suppose that a point $P$ is located in the interior of the pentagon such that $A P = A E$ and $B P = B C$ . Prove that $P$ lies on the diagonal $C E$ if and only if $\operatorname{area}(B C D) + \operatorname{area}(A D E) = \operatorname{area}(A B D) + \operatorname{area}(A B P)$ .

Geometry Solution and answer checking →

Answer key — Stage 10 · Mixed

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