Navvye/Olympiad-Problems-CSV huggingface dataset:
2021 Romanian Masters In Mathematics
Xenia and Sergey play the following game. Xenia thinks of a positive integer $N$ not exceeding $5000$. Then she fixes $20$ distinct positive integers $a_1, a_2, \cdots, a_{20}$ such that, for each $k = 1,2,\cdots,20$, the numbers $N$ and $a_k$ are congruent modulo $k$. By a move, Sergey tells Xenia a set $S$ of positive integers not exceeding $20$, and she tells him back the set $\{a_k : k \in S\}$ without spelling out which number corresponds to which index. How many moves does Sergey need to determine for sure the number Xenia thought of?



Sergey Kudrya, Russia
2021 Romanian Masters In Mathematics
A number of $17$ workers stand in a row. Every contiguous group of at least $2$ workers is a $brigade$. The chief wants to assign each brigade a leader (which is a member of the brigade) so that each worker’s number of assignments is divisible by $4$. Prove that the number of such ways to assign the leaders is divisible by $17$.

$Mikhail$ $Antipov, Russia$
2021 Romanian Masters In Mathematics
Consider an integer \(n \ge 2\) and write the numbers \(1, 2,  \ldots, n\) down on a board. A move consists in erasing any two numbers \(a\) and \(b\), then writing down the numbers \(a+b\) and \(\vert a-b \vert\) on the board, and then removing repetitions (e.g., if the board contained the numbers \(2, 5, 7, 8\), then one could choose the numbers \(a = 5\) and \(b = 7\), obtaining the board with numbers \(2, 8, 12\)). For all integers \(n \ge 2\), determine whether it is possible to be left with exactly two numbers on the board after a finite number of moves.



Proposed by China
2021 Romanian Masters In Mathematics
Let \(n\) be a positive integer. The kingdom of Zoomtopia is a convex polygon with integer sides, perimeter \(6n\), and \(60^\circ\) rotational symmetry (that is, there is a point \(O\) such that a \(60^\circ\) rotation about \(O\) maps the polygon to itself). In light of the pandemic, the government of Zoomtopia would like to relocate its \(3n^2+3n+1\) citizens at \(3n^2+3n+1\) points in the kingdom so that every two citizens have a distance of at least \(1\) for proper social distancing. Prove that this is possible. (The kingdom is assumed to contain its boundary.)



Proposed by Ankan Bhattacharya, USA
2021 Romanian Masters In Mathematics
Initially, a non-constant polynomial $S(x)$ with real coefficients is written down on a board. Whenever the board contains a polynomial $P(x)$, not necessarily alone, one can write down on the board any polynomial of the form $P(C + x)$ or $C + P(x)$ where $C$ is a real constant. Moreover, if the board contains two (not necessarily distinct) polynomials $P(x)$ and $Q(x)$, one can write $P(Q(x))$ and $P(x) + Q(x)$ down on the board. No polynomial is ever erased from the board.



Given two sets of real numbers, $A = \{ a_1, a_2, \dots, a_n \}$ and $B = \{ b_1, \dots, b_n \}$, a polynomial $f(x)$ with real coefficients is $(A,B)$-nice if $f(A) = B$, where $f(A) = \{ f(a_i) : i = 1, 2, \dots, n \}$.



Determine all polynomials $S(x)$ that can initially be written down on the board such that, for any two finite sets $A$ and $B$ of real numbers, with $|A| = |B|$, one can produce an $(A,B)$-nice polynomial in a finite number of steps.



Proposed by Navid Safaei, Iran
2020 Romanian Masters In Mathematics
Let $ABC$ be a triangle with a right angle at $C$. Let $I$ be the incentre of triangle $ABC$, and let $D$ be the foot of the altitude from $C$ to $AB$. The incircle $\omega$ of triangle $ABC$ is tangent to sides $BC$, $CA$, and $AB$ at $A_1$, $B_1$, and $C_1$, respectively. Let $E$ and $F$ be the reflections of $C$ in lines $C_1A_1$ and $C_1B_1$, respectively. Let $K$ and $L$ be the reflections of $D$ in lines $C_1A_1$ and $C_1B_1$, respectively.



Prove that the circumcircles of triangles $A_1EI$, $B_1FI$, and $C_1KL$ have a common point.
2020 Romanian Masters In Mathematics
Let $N \geq 2$ be an integer, and let $\mathbf a$ $= (a_1, \ldots, a_N)$ and $\mathbf b$ $= (b_1, \ldots b_N)$ be sequences of non-negative integers. For each integer $i \not \in \{1, \ldots, N\}$, let $a_i = a_k$ and $b_i = b_k$, where $k \in \{1, \ldots, N\}$ is the integer such that $i-k$ is divisible by $n$. We say $\mathbf a$ is $\mathbf b$-harmonic if each $a_i$ equals the following arithmetic mean: $$a_i = \frac{1}{2b_i+1} \sum_{s=-b_i}^{b_i} a_{i+s}.$$Suppose that neither $\mathbf a $ nor $\mathbf b$ is a constant sequence, and that both $\mathbf a$ is $\mathbf b$-harmonic and $\mathbf b$ is $\mathbf a$-harmonic.



Prove that at least $N+1$ of the numbers $a_1, \ldots, a_N,b_1, \ldots, b_N$ are zero.
2020 Romanian Masters In Mathematics
Let $n\ge 3$ be an integer. In a country there are $n$ airports and $n$ airlines operating two-way flights. For each airline, there is an odd integer $m\ge 3$, and $m$ distinct airports $c_1, \dots, c_m$, where the flights offered by the airline are exactly those between the following pairs of airports: $c_1$ and $c_2$; $c_2$ and $c_3$; $\dots$ ; $c_{m-1}$ and $c_m$; $c_m$ and $c_1$.



Prove that there is a closed route consisting of an odd number of flights where no two flights are operated by the same airline.
2020 Romanian Masters In Mathematics
Let $\mathbb N$ be the set of all positive integers. A subset $A$ of $\mathbb N$ is sum-free if, whenever $x$ and $y$ are (not necessarily distinct) members of $A$,  their sum $x+y$ does not belong to $A$. Determine all surjective functions $f:\mathbb N\to\mathbb N$ such that, for each sum-free subset $A$ of $\mathbb N$, the image $\{f(a):a\in A\}$ is also sum-free.



Note: a function $f:\mathbb N\to\mathbb N$ is surjective if, for every positive integer $n$, there exists a positive integer $m$ such that $f(m)=n$.
2020 Romanian Masters In Mathematics
A lattice point in the Cartesian plane is a point whose coordinates are both integers. A lattice polygon is a polygon all of whose vertices are lattice points.



Let $\Gamma$ be a convex lattice polygon. Prove that $\Gamma$ is contained in a convex lattice polygon $\Omega$ such that the vertices of $\Gamma$ all lie on the boundary of $\Omega$, and exactly one vertex of $\Omega$ is not a vertex of $\Gamma$.
2020 Romanian Masters In Mathematics
For each integer $n \geq 2$, let $F(n)$ denote the greatest prime factor of $n$. A strange pair is a pair of distinct primes $p$ and $q$ such that there is no integer $n \geq 2$ for which $F(n)F(n+1)=pq$.



Prove that there exist infinitely many strange pairs.
2019 Romanian Master of Mathematics Shortlist
Determine all the functions $f:\mathbb R\mapsto\mathbb R$ satisfies the equation

$f(a^2 +ab+ f(b^2))=af(b)+b^2+ f(a^2)\,\forall a,b\in\mathbb R $
2019 Romanian Master of Mathematics Shortlist
Let $k$ and $N$ be integers such that $k > 1$ and $N > 2k + 1$. A number of $N$ persons sit around the Round Table, equally spaced. Each person is either a knight (always telling the truth) or a liar (who always lies). Each person sees the nearest k persons clockwise, and the nearest $k$ persons anticlockwise. Each person says: ''I see equally many knights to my left and to my right." Establish, in terms of $k$ and $N$, whether the persons around the Table are necessarily all knights.



Sergey Berlov, Russia
2019 Romanian Master of Mathematics Shortlist
Fix an integer $n \ge 2$. A fairy chess piece leopard  may move one cell up, or one cell to the right, or one cell diagonally down-left. A leopard is placed onto some cell of a $3n \times 3n$ chequer board. The leopard makes several moves, never visiting a cell twice, and comes back to the starting cell. Determine the largest possible number of moves the leopard could have made.



Dmitry Khramtsov, Russia
2019 Romanian Master of Mathematics Shortlist
Fix an odd integer $n > 1$. For a permutation $p$ of the set $\{1,2,...,n\}$, let S be the number of pairs of indices $(i, j)$, $1 \le i \le  j \le  n$, for which $p_i +p_{i+1} +...+p_j$ is divisible by $n$. Determine the maximum possible value of $S$.



Croatia
2019 Romanian Master of Mathematics Shortlist
Let $BM$ be a median in an acute-angled triangle $ABC$. A point $K$ is chosen on the line through $C$ tangent to the circumcircle of $\vartriangle BMC$ so that $\angle KBC = 90^\circ$. The segments $AK$ and $BM$ meet at $J$. Prove that the circumcenter of  $\triangle BJK$ lies on the line $AC$.



Aleksandr Kuznetsov, Russia
2019 Romanian Master of Mathematics Shortlist
Let $ABC$ be an acute-angled triangle. The line through $C$ perpendicular to $AC$ meets the external angle bisector of $\angle ABC$ at $D$. Let $H$ be the foot of the perpendicular from $D$ onto $BC$. The point $K$ is chosen on $AB$ so that $KH \parallel AC$. Let $M$ be the midpoint of $AK$. Prove that $MC = MB + BH$.



Giorgi Arabidze, Georgia,
2019 Romanian Master of Mathematics Shortlist
Let $ABC$ be an acute-angled triangle with $AB \ne AC$, and let $I$ and $O$ be its incenter and circumcenter, respectively. Let the incircle touch $BC, CA$ and $AB$ at $D, E$ and $F$, respectively. Assume that the line through $I$ parallel to $EF$, the line through $D$ parallel to$ AO$, and the altitude from $A$ are concurrent. Prove that the concurrency point is the orthocenter of the triangle $ABC$.



Petar Nizic-Nikolac, Croatia
2019 Romanian Master of Mathematics Shortlist
Let  $\Omega$ be the circumcircle of an acute-angled triangle $ABC$. Let $D$ be the midpoint of the minor arc $AB$ of $\Omega$. A circle  $\omega$ centered at $D$ is tangent to $AB$ at $E$. The tangents to $\omega$ through $C$ meet the segment $AB$ at $K$ and $L$, where $K$ lies on the segment $AL$. A circle  $\Omega_1$ is tangent to the segments $AL, CL$, and also to $ \Omega$  at point $M$. Similarly, a circle  $\Omega_2$ is tangent to the segments $BK, CK$, and also to  $\Omega$ at point $N$. The lines $LM$ and $KN$ meet at $P$. Prove that $\angle KCE = \angle LCP$.



Poland
2019 Romanian Master of Mathematics Shortlist
Let $\Omega$  be the circumcircle of an acute-angled triangle $ABC$. A point $D$ is chosen on the internal bisector of $\angle ACB$ so that the points $D$ and $C$ are separated by $AB$. A circle $\omega$ centered at $D$ is tangent to the segment $AB$ at $E$. The tangents to $\omega$ through $C$ meet the segment $AB$ at $K$ and $L$, where $K$ lies on the segment $AL$. A circle  $\Omega_1$ is tangent to the segments $AL, CL$, and also to $\Omega$ at point $M$. Similarly, a circle $\Omega_2$ is tangent to the segments $BK, CK$, and also to  $\Omega$ at point $N$. The lines $LM$ and $KN$ meet at $P$. Prove that $\angle KCE = \angle LCP$.



Poland
2019 Romanian Master of Mathematics Shortlist
A quadrilateral $ABCD$ is circumscribed about a circle with center $I$. A point $P \ne I$ is chosen inside $ABCD$ so that the triangles $PAB, PBC, PCD,$ and $PDA$ have equal perimeters. A circle $\Gamma$ centered at $P$ meets the rays $PA, PB, PC$, and $PD$ at $A_1, B_1, C_1$, and $D_1$, respectively. Prove that the lines $PI, A_1C_1$, and $B_1D_1$ are concurrent.



Ankan Bhattacharya, USA
2019 Romanian Master of Mathematics Shortlist
Let $p$ and $q$ be relatively prime positive odd integers such that $1 < p < q$. Let $A$ be a set of pairs of integers $(a, b)$, where $0 \le a \le p - 1, 0 \le b \le q - 1$, containing exactly one pair from each of the sets $$\{(a, b),(a + 1, b + 1)\},  \{(a, q - 1), (a + 1, 0)\}, \{(p - 1,b),(0, b + 1)\}$$whenever $0 \le a \le p - 2$ and $0 \le b \le q - 2$. Show that $A$ contains at least $(p - 1)(q + 1)/8$ pairs whose entries are both even.



Agnijo Banerjee and Joe Benton, United Kingdom
2019 Romanian Master of Mathematics Shortlist
Let there be an equilateral triangle $ABC$ and a point $P$ in its plane such that $AP<BP<CP.$  Suppose that the lengths of segments $AP,BP$ and $CP$ uniquely determine the side of $ABC$. Prove that $P$ lies on the circumcircle of triangle $ABC.$
2019 Romanian Master of Mathematics Shortlist
Two ants are moving along the edges of a convex polyhedron. The route of every ant ends in its starting point, so that one ant does not pass through the same point twice along its way. On every face $F$ of the polyhedron are written the number of edges of $F$ belonging to the route of the first ant and the number of edges of $F$ belonging to the route of the second ant. Is there a polyhedron and a pair of routes described as above, such that only one face contains a pair of distinct numbers?





Proposed by Nikolai Beluhov
2019 Romanian Master of Mathematics Shortlist
Let $P(x)$ be a nonconstant complex coefficient polynomial and let $Q(x,y)=P(x)-P(y).$ Suppose that polynomial $Q(x,y)$ has exactly $k$ linear factors unproportional two by tow (without counting repetitons). Let $R(x,y)$ be factor of $Q(x,y)$ of degree strictly smaller than $k$. Prove that $R(x,y)$ is a product of linear polynomials.



Note:  The degree of nontrivial polynomial $\sum_{m}\sum_{n}c_{m,n}x^{m}y^{n}$ is the maximum of $m+n$ along all nonzero coefficients $c_{m,n}.$ Two polynomials are proportional if one of them is the other times a complex constant.







Proposed by Navid Safaie
2019 Romanian Masters In Mathematics
Amy and Bob play the game. At the beginning, Amy writes down a positive integer on the board. Then the players take moves in turn, Bob moves first. On any move of his, Bob replaces the number $n$ on the blackboard with a number of the form $n-a^2$, where $a$ is a positive integer. On any move of hers, Amy replaces the number $n$ on the blackboard with a number of the form $n^k$, where $k$ is a positive integer. Bob wins if the number on the board becomes zero.

Can Amy prevent Bob’s win?



Maxim Didin, Russia
2019 Romanian Masters In Mathematics
Let $ABCD$ be an isosceles trapezoid with $AB\parallel CD$. Let $E$ be the midpoint of $AC$. Denote by $\omega$ and $\Omega$ the circumcircles of the triangles $ABE$ and $CDE$, respectively. Let $P$ be the crossing point of the tangent to $\omega$ at $A$ with the tangent to $\Omega$ at $D$. Prove that $PE$ is tangent to $\Omega$.



Jakob Jurij Snoj, Slovenia
2019 Romanian Masters In Mathematics
Given any positive real number $\varepsilon$, prove that, for all but finitely many positive integers $v$, any graph on $v$ vertices with at least  $(1+\varepsilon)v$ edges has two distinct simple cycles of equal lengths.

(Recall that the notion of a simple cycle does not allow repetition of vertices in a cycle.)



Fedor Petrov, Russia
2019 Romanian Masters In Mathematics
Prove that for every positive integer $n$ there exists a (not necessarily convex) polygon with no three collinear vertices, which admits exactly $n$ diffferent triangulations.



(A triangulation is a dissection of the polygon into triangles by interior diagonals which have no common interior points with each other nor with the sides of the polygon)
2019 Romanian Masters In Mathematics
Determine all functions $f: \mathbb{R} \to \mathbb{R}$ satisfying

$$f(x + yf(x)) + f(xy) = f(x) + f(2019y),$$for all real numbers $x$ and $y$.
2019 Romanian Masters In Mathematics
Find all pairs of integers $(c, d)$, both greater than 1, such that the following holds:



For any monic polynomial $Q$ of degree $d$ with integer coefficients and for any prime $p > c(2c+1)$, there exists a set $S$ of at most $\big(\tfrac{2c-1}{2c+1}\big)p$ integers, such that

$$\bigcup_{s \in S} \{s,\; Q(s),\; Q(Q(s)),\; Q(Q(Q(s))),\; \dots\}$$contains a complete residue system modulo $p$ (i.e., intersects with every residue class modulo $p$).
2018 Romanian Master of Mathematics Shortlist
Let $m$ and $n$ be integers greater than $2$, and let $A$ and $B$ be non-constant polynomials with complex coefficients, at least one of which has a degree greater than $1$. Prove that if the degree of the polynomial $A^m-B^n$ is less than $\min(m,n)$, then $A^m=B^n$.



Proposed by Tobi Moektijono, Indonesia
2018 Romanian Master of Mathematics Shortlist
Call a point in the Cartesian plane with integer coordinates a $lattice$ $point$. Given a finite set $\mathcal{S}$ of lattice points we repeatedly perform the following operation: given two distinct lattice points $A, B$ in $\mathcal{S}$ and two distinct lattice points $C, D$ not in $\mathcal{S}$ such that $ACBD$ is a parallelogram with $AB > CD$, we replace $A, B$ by $C, D$. Show that only finitely many such operations can be performed.



Proposed by Joe Benton, United Kingdom.
2018 Romanian Master of Mathematics Shortlist
Fix integers $n\ge k\ge 2$. We call a collection of integral valued coins $n-diverse$ if no value occurs in it more than $n$ times. Given such a collection, a number $S$ is $n-reachable$ if that collection contains $n$ coins whose sum of values equals $S$. Find the least positive integer $D$ such that for any $n$-diverse collection of $D$ coins there are at least $k$ numbers that are $n$-reachable.



Proposed by Alexandar Ivanov, Bulgaria.
2018 Romanian Master of Mathematics Shortlist
$N$ teams take part in a league. Every team plays every other team exactly once during the league, and receives 2 points for each win, 1 point for each draw, and 0 points for each loss. At the end of the league, the sequence of total points in descending order $\mathcal{A} = (a_1 \ge a_2 \ge \cdots \ge a_N )$ is known, as well as which team obtained which score. Find the number of sequences $\mathcal{A}$ such that the outcome of all matches is uniquely determined by this information.



Proposed by Dominic Yeo, United Kingdom.
2018 Romanian Master of Mathematics Shortlist
Let $k$ and $s$ be positive integers such that $s<(2k + 1)^2$. Initially, one cell out of an $n \times n$ grid is coloured green. On each turn, we pick some green cell $c$ and colour green some $s$ out of the $(2k + 1)^2$ cells in the $(2k + 1) \times (2k + 1)$ square centred at $c$. No cell may be coloured green twice. We say that $s$ is $k-sparse$ if there exists some positive number $C$ such that, for every positive integer $n$, the total number of green cells after any number of turns is always going to be at most $Cn$. Find, in terms of $k$, the least $k$-sparse integer $s$.



Proposed by Nikolai Beluhov.
2018 Romanian Master of Mathematics Shortlist
Let $ABC$ be a triangle and let $H$ be the orthogonal projection of $A$ on the line $BC$. Let $K$ be a point on the segment $AH$ such that $AH = 3 KH$. Let $O$ be the circumcenter of triangle $ABC$ and let $M$ and $N$ be the midpoints of sides $AC$ and $AB$ respectively. The lines $KO$ and $MN$ meet at a point $Z$ and the perpendicular at $Z$ to $OK$ meets lines $AB, AC$ at $X$ and $Y$ respectively. Show that $\angle XKY = \angle CKB$.



Italy
2018 Romanian Master of Mathematics Shortlist
Let $\triangle ABC$ be a triangle, and let $S$ and $T$ be the midpoints of the sides $BC$ and $CA$, respectively. Suppose $M$ is the midpoint of the segment $ST$ and the circle $\omega$ through $A, M$ and $T$ meets the line $AB$ again at $N$. The tangents of $\omega$ at $M$ and $N$ meet at $P$. Prove that $P$ lies on $BC$ if and only if the triangle $ABC$ is isosceles with apex at $A$.



Proposed by Reza Kumara, Indonesia
2018 Romanian Master of Mathematics Shortlist
Determine all polynomials $f$ with integer coefficients such that $f(p)$ is a divisor of $2^p-2$ for every odd prime $p$.



Proposed by Italy
2018 Romanian Master of Mathematics Shortlist
Prove that for each positive integer $k$ there exists a number base $b$ along with $k$ triples of Fibonacci numbers $(F_u,F_v,F_w)$ such that when they are written in base $b$, their concatenation is also a Fibonacci number written in base $b$. (Fibonacci numbers are defined by $F_1 = F_2 = 1$ and $F_{n+2} = F_{n+1} + F_n$ for all positive integers $n$.)





Proposed by Serbia
2018 Romanian Masters in Mathematics
Let $ABCD$ be a cyclic quadrilateral an let $P$ be a point on the side $AB.$ The diagonals $AC$ meets the segments $DP$ at $Q.$ The line through $P$ parallel to $CD$ mmets the extension of the side $CB$ beyond $B$ at $K.$ The line through $Q$ parallel to $BD$ meets the extension of the side $CB$ beyond $B$ at $L.$ Prove that the circumcircles of the triangles $BKP$ and $CLQ$ are tangent .
2018 Romanian Masters in Mathematics
Determine whether there exist non-constant polynomials $P(x)$ and $Q(x)$ with real coefficients satisfying

$$P(x)^{10}+P(x)^9 = Q(x)^{21}+Q(x)^{20}.$$
2018 Romanian Masters in Mathematics
Ann and Bob play a game on the edges of an infinite square grid, playing in turns. Ann plays the first move. A move consists of orienting any edge that has not yet been given an orientation. Bob wins if at any point a cycle has been created. Does Bob have a winning strategy?
2018 Romanian Masters in Mathematics
Let $a,b,c,d$ be positive integers such that $ad \neq bc$ and $gcd(a,b,c,d)=1$. Let $S$ be the set of values attained by $\gcd(an+b,cn+d)$ as $n$ runs through the positive integers. Show that $S$ is the set of all positive divisors of some positive integer.
2018 Romanian Masters in Mathematics
Let $n$ be positive integer and fix $2n$ distinct points on a circle. Determine the number of ways to connect the points with $n$ arrows (oriented line segments) such that all of the following conditions hold:



each of the $2n$ points is a startpoint or endpoint 	of an arrow;

no two arrows intersect; and

there are no two arrows $\overrightarrow{AB}$ and 	$\overrightarrow{CD}$ such that $A$, $B$, $C$ and $D$ 	appear in clockwise order around the circle 	(not necessarily consecutively).
2018 Romanian Masters in Mathematics
Fix a circle $\Gamma$, a line $\ell$ to tangent $\Gamma$, and another circle $\Omega$ disjoint from $\ell$ such that $\Gamma$ and $\Omega$ lie on opposite sides of $\ell$. The tangents to $\Gamma$ from a variable point $X$ on $\Omega$ meet $\ell$ at $Y$ and $Z$. Prove that, as $X$ varies over $\Omega$, the circumcircle of $XYZ$ is tangent to two fixed circles.
2017 Romanian Master of Mathematics Shortlist
A set $A$ is endowed with a binary operation $*$ satisfying the following four conditions:

(1) If $a, b, c$ are elements of $A$, then $a * (b * c) = (a * b) * c$ ,

(2) If $a, b, c$ are elements of $A$ such that $a * c = b *c$, then $a = b$ ,

(3) There exists an element $e$ of $A$ such that $a * e = a$ for all $a$ in $A$, and

(4) If a and b are distinct elements of $A-\{e\}$, then $a^3 * b = b^3 * a^2$, where $x^k = x * x^{k-1}$ for all integers $k \ge 2$ and all $x$ in $A$.

Determine the largest cardinality $A$ may have.



proposed by Bojan Basic, Serbia
2017 Romanian Master of Mathematics Shortlist
A planar country has an odd number of cities separated by pairwise distinct distances. Some of these cities are connected by direct two-way flights. Each city is directly connected to exactly two ther cities, and the latter are located farthest from it. Prove that, using these flights, one may go from any city to any other city
2017 Romanian Master of Mathematics Shortlist
Fix an integer $n \ge 2$ and let $A$ be an $n\times n$ array with $n$ cells cut out so that exactly one cell is removed out of every row and every column. A stick  is a $1\times k$ or $k\times 1$ subarray of $A$, where $k$ is a suitable positive integer.

(a) Determine the minimal number of sticks  $A$ can be dissected into.

(b) Show that the number of ways to dissect $A$ into a minimal number of sticks  does not exceed $100^n$.



proposed by Palmer Mebane and Nikolai Beluhov



a few commentsa variation of part a, was problem 5

a  variation of part b, was posted here

this post was made in order to complete the post collection of RMM Shortlist 2017
2017 Romanian Master of Mathematics Shortlist
Let $ABCD$ be a trapezium, $AD\parallel BC$, and let $E,F$ be points on the sides$AB$ and $CD$, respectively. The circumcircle of $AEF$ meets $AD$ again at $A_1$, and the circumcircle of $CEF$ meets $BC$ again at $C_1$. Prove that $A_1C_1,BD,EF$ are concurrent.
2017 Romanian Master of Mathematics Shortlist
Let $ABC$ be a triangle. Consider the circle $\omega_B$ internally tangent to the sides $BC$ and $BA$, and to the circumcircle of the triangle $ABC$, let $P$ be the point of contact of the two circles. Similarly, consider the circle $\omega_C$ internally tangent to the sides $CB$ and $CA$, and to the circumcircle of the triangle $ABC$, let $Q$ be the point of contact of the two circles. Show that the incentre of the triangle $ABC$ lies on the segment $PQ$ if and only if $AB + AC = 3BC$.



proposed by Luis Eduardo Garcia Hernandez, Mexico