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IMO 2019 – Problems

July 18, 2019 beni22sof Leave a comment Go to comments

Problem 1. Let ${\Bbb{Z}}$ be the set of integers. Determine all functions ${f:\Bbb Z \rightarrow \Bbb Z}$ such that, for all integers ${a}$ and ${b}$ ,

$\displaystyle f(2a)+2f(b) = f(f(a+b)).$

Problem 2. In triangle ${ABC}$ , point ${A_1}$ lies on the side ${BC}$ and point ${B_1}$ lies on side ${AC}$ . Let ${P}$ and ${Q}$ be points on segments ${AA_1}$ and ${BB_1}$ , respectively, such that ${PQ}$ is parallel to ${AB}$ . Let ${P_1}$ be a point on line ${PB_1}$ such that ${B_1}$ lies strictly between ${P}$ and ${P_1}$ and ${\angle PP_1C=\angle BAC}$ . Similarly, let ${Q_1}$ be a point on line ${QA_1}$ such that ${A_1}$ lies strictly between ${Q}$ and ${Q_1}$ and ${\angle CQ_1Q = \angle CBA}$ .

Prove that points ${P,Q,P_1}$ and ${Q_1}$ are concyclic.

Problem 3. A social network has ${2019}$ users, some pairs of whom are friends. Whenever user ${A}$ is friends with user ${B}$ , user ${B}$ is also friends with user ${A}$ . Events of the following kind may happen repeatedly, one at a time:

Three users ${A,B,C}$ such that ${A}$ is friends with both ${B}$ and ${C}$ but ${B}$ and ${C}$ are not friends change their friendship statuses such that ${B}$ and ${C}$ are now friends, but ${A}$ is no longer friends with ${B}$ and no longer friends with ${C}$ . All other friendship statuses are unchanged.

Initially, ${1010}$ users have ${1009}$ friends each and ${1009}$ users haf ${1010}$ friends each. Prove that there exists a sequence of such events after which each user is friends with at most one other user.

Problem 4. Find all pairs ${(k,n)}$ of positive integers such that

$\displaystyle k! = (2^n-1)(2^n-2)(2^n-4)...(2^n-2^{n-1}).$

Problem 5. The Bank of Bath issues coins with an ${H}$ on a side and a ${T}$ on the other. Harry has ${n}$ of these coins arranged in a line from left to right. He repeatedly performs the following operation: if there are exactly ${k>0}$ coins showing ${H}$ , then he turns over the ${k}$ th coin from the left; otherwise, all coins show ${T}$ and he stops. For example, if ${n=3}$ the process starting with the configuration ${THT}$ would be

$\displaystyle THT \rightarrow HHT \rightarrow HTT \rightarrow TTT,$

which stops after three operations.

(a) Show that, for each initial configuration, Harry stops after a finite number of operations.
(b) For each initial configuration ${C}$ , let ${L(C)}$ be the number of operation before Harry stops. For example ${L(THT)=3}$ and ${L(TTT)=0}$ . Determine the average value of ${L(C)}$ over all ${2^n}$ possible initial configurations ${C}$ .

Problem 6. Let ${I}$ be the incenter of the acute triangle ${ABC}$ with ${AB \neq AC}$ . The incircle ${\omega}$ of ${ABC}$ is tangent to sides ${BC,CA}$ and ${AB}$ at ${D,E,F}$ , respectively. The line through ${D}$ perpendicular to ${EF}$ meets ${\omega}$ again at ${R}$ . The line ${AR}$ meets ${\omega}$ again at ${P}$ . THe circumcircles of triangles ${PCE}$ and ${PBF}$ meet again at ${Q}$ .