Navvye/Olympiad-Problems-CSV huggingface dataset:
2016 Iranian Geometry Olympiad
Let the circles $\omega$ and $\omega^ \prime$ intersect in $A$ and $B$. Tangent to circle$\omega$ at $A$ intersects$\omega^ \prime$ in $C$ and tangent to circle $\omega^ \prime$ at $A$ intersects $\omega$ in $D$. Suppose that $CD$ intersects$\omega$ and $\omega^ \prime$ in $E$ and $F$, respectively (assume that $E$ is between $F$ and $C$). The perpendicular to $AC$ from $E$ intersects $\omega^ \prime$ in point $P$ and perpendicular to $AD$ from $F$ intersects$\omega$ in point $Q$ (The points $A, P$ and $Q$ lie on the same side of the line $CD$). Prove that the points $A, P$ and $Q$ are collinear.

Proposed by Mahdi Etesami Fard
2016 Iranian Geometry Olympiad
In acute-angled triangle $ABC$, altitude of $A$ meets $BC$ at $D$, and $M$ is midpoint of $AC$. Suppose that $X$ is a point such that $\measuredangle AXB = \measuredangle DXM =90^\circ$ (assume that $X$ and $C$ lie on opposite sides of the line $BM$). Show that $\measuredangle XMB = 2\measuredangle MBC$.Proposed by Davood Vakili
2016 Iranian Geometry Olympiad
In a convex qualrilateral $ABCD$, let $P$ be the intersection point of $AD$ and $BC$. Suppose that $I_1$ and $I_2$ are the incenters of triangles $PAB$ and $PDC$,respectively. Let $O$ be the circumcenter of $PAB$, and $H$ the orthocenter of $PDC$. Show that the circumcircles of triangles $AI_1B$ and $DHC$ are tangent together if and only if the circumcircles of triangles $AOB$ and $DI_2C$ are tangent together.

Proposed by Hooman Fattahimoghaddam
2016 Iranian Geometry Olympiad
In a convex quadrilateral $ABCD$, the lines $AB$ and $CD$ meet at point $E$ and the lines $AD$ and $BC$ meet at point $F$. Let $P$ be the intersection point of diagonals $AC$ and $BD$. Suppose that $\omega_1$ is a circle passing through $D$ and tangent to $AC$ at $P$. Also suppose that $\omega_2$ is a circle passing through $C$ and tangent to $BD$ at $P$. Let $X$ be the intersection point of $\omega_1$ and $AD$, and $Y$ be the intersection point of $\omega_2$ and $BC$. Suppose that the circles $\omega_1$ and $\omega_2$ intersect each other in $Q$ for the second time. Prove that the perpendicular from $P$ to the line $EF$ passes through the circumcenter of triangle $XQY$ .

Proposed by Iman Maghsoudi
2016 Iranian Geometry Olympiad
Do there exist six points $X_1,X_2,Y_1, Y_2,Z_1,Z_2$ in the plane such that all of the triangles $X_iY_jZ_k$ are similar for $1\leq i, j, k \leq 2$?

Proposed by Morteza Saghafian
2015 Iran Geometry Olympiad
We have four wooden triangles with sides $3, 4, 5$ centimeters. How many convex polygons can we make by all of these triangles? (Just draw the polygons without any proof)



A convex polygon is a polygon which all of it's angles are less than $180^o$ and there isn't any hole in it. For example:

//cdn.artofproblemsolving.com/images/7/5/4/7545261bf092f2c843dcb006847dda80e25acedc.png
2015 Iran Geometry Olympiad
Let ABC be a triangle with $\angle A = 60^o$. The points $M,N,K$ lie on $BC,AC,AB$ respectively such that $BK = KM = MN = NC$. If $AN = 2AK$, find the values of $\angle B$ and $\angle C$.



by Mahdi Etesami Fard
2015 Iran Geometry Olympiad
In the figure below, we know that $AB = CD$ and $BC = 2AD$. Prove that $<BAD = 30^o$.

//cdn.artofproblemsolving.com/images/d/d/4/dd4b270f8196cb680016b69bfae8112ab98ea492.png
2015 Iran Geometry Olympiad
In rectangle $ABCD$, the points $M,N,P,Q$ lie on $AB,BC,CD,DA$ respectively such that the area of triangles $AQM,BMN,CNP,DPQ$ are equal. Prove that the quadrilateral $MNPQ$ is parallelogram.



by Mahdi Etesami Fard
2015 Iran Geometry Olympiad
Do there exist $6$ circles in the plane such that every circle passes through centers of exactly $3$ other circles?



by Morteza Saghafian
2015 Iran Geometry Olympiad
Given a circle and Points $P,B,A$ on it.Point $Q$ is Interior of this circle such that:

$1)$ $\angle PAQ=90$.

$ 2)PQ=BQ$.

Prove that $\angle AQB - \angle PQA=\stackrel{\frown}{AB}$.



proposed by Davoud Vakili, Iran.
2015 Iran Geometry Olympiad
In acute-angled triangle $ABC$, $BH$ is the altitude of the vertex $B$. The points $D$ and $E$ are midpoints of $AB$ and $AC$ respectively. Suppose that $F$ be the reflection of $H$ with respect to $ED$. Prove that the line $BF$ passes through circumcenter of $ABC$.



by Davood Vakili
2015 Iran Geometry Olympiad
In triangle $ABC$ ,$M,N,K$ are midpoints of sides $BC,AC,AB$,respectively.Construct two semicircles with diameter $AB,AC$ outside of triangle $ABC$.$MK,MN$ intersect with semicircles in $X,Y$.The tangents to semicircles at $X,Y$ intersect at point $Z$.Prove that $AZ \perp BC$.(Mehdi E'tesami Fard)
2015 Iran Geometry Olympiad
a) Do there exist 5 circles in the plane such that every circle passes through centers

of exactly 3 circles?

b) Do there exist 6 circles in the plane such that every circle passes through centers

of exactly 3 circles?
2015 Iran Geometry Olympiad
let $ w_1 $ and $ w_2 $ two circles such that $ w_1 \cap w_2 = \{ A , B \} $



let $ X $ a point on $ w_2 $ and $ Y $ on $ w_1 $ such that $ BY \bot BX $



suppose that $ O $ is the center of $ w_1 $ and $ X' = w_2 \cap OX $



now if $ K = w_2 \cap X'Y $ prove $ X $ is the midpoint of arc $ AK $
2015 Iran Geometry Olympiad
let $ ABC $ an equilateral triangle with circum circle $ w $



let $ P $ a point on arc $ BC $ ( point $ A $ is on the other side )



pass a tangent line $ d $ through point $ P $ such that $ P \cap AB = F $ and $ AC \cap d = L $



let $ O $ the center of the circle $ w $



prove that $ \angle LOF > 90^{0} $
2015 Iran Geometry Olympiad
let $ H $ the orthocenter of the triangle $ ABC $



pass two lines $ l_1 $ and $ l_2 $ through $ H $ such that $ l_1 \bot l_2 $



we have $ l_1 \cap BC = D $ and $ l_1 \cap AB = Z $



also $ l_2 \cap BC = E $ and $ l_2 \cap AC = X $ like this picture



pass a line $ d_1$ through $ D $ parallel to $ AC $ and another line $ d_2 $ through $ E $ parallel to $ AB $



let $ d_1 \cap d_2 = Y $



prove $ X $ $ , $ $ Y $ and $ Z $ are on a same line
2015 Iran Geometry Olympiad
In triangle $ABC$, we draw the circle with center $A$ and radius $AB$. This circle intersects $AC$ at two points. Also we draw the circle with center $A$ and radius $AC$ and this circle intersects $AB$ at two points. Denote these four points by $A_1, A_2, A_3, A_4$. Find the points $B_1, B_2, B_3, B_4$ and $C_1, C_2, C_3, C_4$ similarly. Suppose that these $12$ points lie on two circles. Prove that the triangle $ABC$ is isosceles.
2015 Iran Geometry Olympiad
we have  a triangle $ ABC $ and make rectangles $ ABA_1B_2 $ , $ BCB_1C_2 $ and $ CAC_1A_2 $ out of it.



then pass a line through $ A_2 $ perpendicular to $ C_1A_2 $ and pass another line through $ A_1 $ perpendicular to $  A_1B_2 $.



let $ A' $ the common point of this two lines.

like this we make $ B' $ and $ C' $.



prove $ AA' $ , $ BB' $ and $ CC' $ intersect each other in a same point.
2014 Iran Geometry Olympiad (senior)




ATimo wrote:

ABC is a triangle with A=90 and C=30.Let M be the midpoint of BC. Let W be a circle passing through A tangent in M to BC. Let P be the circumcircle of ABC. W is intersecting AC in N and P in M. prove that MN is perpendicular to BC.



Let me correct your question and $\text{\LaTeX}\text{ify}$ it. :)



Corrected QuestionQuestion

$ABC$ is a triangle with $\angle BAC=90^{\circ}$ and $\angle ACB=30^{\circ}$. Let $M_1$ be the midpoint of $BC$. Let $W$ be a circle passing through $A$ tangent in $M_1$ to $BC$. Let $P$ be the circumcircle of $ABC$. $W$ is intersecting $AC$ in $N$ and $P$ in $M$. Prove that $MN$ is perpendicular to $BC$.



SolutionSolution

Let $M_1$ be the midpoint of $BC$. And let $C_1$ be the center of $W$. So, we first show that $C_1\in AC$.

Clearly $C_1M_1\perp BC$. And also, since $M_1$ is the center of $P$ as $ABC$ is a right triangle, we have $M_1A=M_1B=M_1C$.

So, $\angle M_1AB=\angle M_1BA=60^{\circ}$. And this implies that $\triangle AM_1B$ is equilateral. So, $\angle C_1M_1A=90^{\circ}-60^{\circ}=30^{\circ}=\angle C_1AM_1$.

But then, $\angle C_1AM_1+\angle M_1AB=90^{\circ}=\angle C_1AB=\angle CAB$. This shows that $\boxed{C_1\in AC}$.

Next $C_1N=C_1M_1$ and $\angle AC_1M_1=180^{\circ}-30^{\circ}-30^{\circ}=120^{\circ}=180^{\circ}-\angle NC_1M_1$, showing that $\triangle NC_1M_1$ is equilateral.

Again, $C_1M=C_1A$ showing $\angle C_1AM=\angle C_1MA$.

Now note that $AM$ is the radical axis of $P$ and $W$. So, $M_1C_1\perp AM$. Let $M_1C_1\cap AM=X$. Then,

$AC_1X=60^{\circ}\Rightarrow NC_1M=180^{\circ}-\angle MC_1X-60^{\circ}=60^{\circ}$.

But $C_1M=C_1N$. Thus $\angle MNC_1=\angle NC_1M\Rightarrow MN\parallel M_1C_1\Rightarrow \boxed{MN\perp BC}$.

This completes the proof. :)
2014 Iran Geometry Olympiad (senior)
In the Quadrilateral $ABCD$ we have $ \measuredangle B=\measuredangle D = 60^\circ $.$M$ is midpoint of side $AD$.The line through $M$ parallel to $CD$ meets $BC$ at $P$.Point $X$ lying on $CD$ such that $BX=MX$.Prove that $AB=BP$ if and only if $\measuredangle MXB=60^\circ$.



Author: Davoud Vakili, Iran
2014 Iran Geometry Olympiad (senior)
Let $ABC$ be an acute triangle.A circle with diameter $BC$ meets $AB$ and $AC$ at $E$ and $F$,respectively. $M$ is midpoint of $BC$ and $P$ is point of intersection $AM$ with $EF$. $X$ is an arbitary point on arc $EF$ and $Y$ is the second intersection of $XP$ with a circle with diameter $BC$.Prove that $ \measuredangle XAY=\measuredangle XYM $.

Author:Ali zo'alam , Iran
2014 Iran Geometry Olympiad (senior)
A tangent line to circumcircle of acute triangle $ABC$ ($AC>AB$) at $A$ intersects with the extension of $BC$ at $P$. $O$ is the circumcenter of triangle $ABC$.Point $X$ lying on $OP$ such that $\measuredangle AXP=90^\circ$.Points $E$ and $F$ lying on $AB$ and $AC$,respectively,and they are in one side of line $OP$ such that $ \measuredangle EXP=\measuredangle ACX $  and $\measuredangle FXO=\measuredangle ABX $.

$K$,$L$ are points of intersection $EF$ with circumcircle of triangle $ABC$.prove that $OP$ is tangent to circumcircle of triangle $KLX$.



Author:Mehdi E'tesami Fard , Iran
2014 Iran Geometry Olympiad (senior)
Two points $P$ and $Q$ lying on side $BC$ of  triangle $ABC$ and their distance from the midpoint of $BC$ are equal.The perpendiculars from $P$ and $Q$ to $BC$ intersect $AC$ and $AB$ at $E$ and $F$,respectively.$M$ is point of intersection $PF$ and $EQ$.If $H_1$ and $H_2$ be the orthocenters of triangles $BFP$ and $CEQ$, respectively, prove that $ AM\perp H_1H_2 $.



Author:Mehdi E'tesami Fard , Iran
2014 Iranian Geometry Olympiad (junior)
ABC is a triangle with A=90 and C=30.Let M be the midpoint of BC. Let W be a circle passing through A tangent in M to BC. Let P be the circumcircle of ABC. W is intersecting AC in N and P in M. prove that MN is perpendicular to BC.
2014 Iranian Geometry Olympiad (junior)
The inscribed circle of $\triangle ABC$ touches $BC, AC$ and $AB$ at $D,E$ and $F$ respectively. Denote the perpendicular foots from $F, E$ to $BC$ by $K, L$ respectively. Let the second intersection of these perpendiculars with the incircle be $M, N$ respectively. Show that $\frac{{{S}_{\triangle BMD}}}{{{S}_{\triangle CND}}}=\frac{DK}{DL}$



by Mahdi Etesami Fard
2014 Iranian Geometry Olympiad (junior)
Each of Mahdi and Morteza has drawn an inscribed $93$-gon. Denote the first one by $A_1A_2…A_{93}$ and the second by $B_1B_2…B_{93}$. It is known that $A_iA_{i+1} // B_iB_{i+1}$ for $1 \le i \le 93$ ($A_{93} = A_1, B_{93} = B_1$). Show that $\frac{A_iA_{i+1} }{ B_iB_{i+1}}$ is a constant number independent of $i$.



by Morteza Saghafian
2014 Iranian Geometry Olympiad (junior)
In a triangle ABC we have $\angle C = \angle A + 90^o$. The point $D$ on the continuation of $BC$ is given such that $AC = AD$. A point $E$ in the side of $BC$ in which $A$ doesn’t lie is chosen such that $\angle EBC = \angle A, \angle EDC = \frac{1}{2} \angle A$ . Prove that $\angle CED =  \angle ABC$.



by Morteza Saghafian
2014 Iranian Geometry Olympiad (junior)
Two points $X, Y$ lie on the arc $BC$ of the circumcircle of $\triangle ABC$ (this arc does not contain $A$) such that $\angle BAX = \angle CAY$ . Let $M$ denotes the midpoint of the chord $AX$ . Show that   $BM +CM > AY$ .



by Mahan Tajrobekar
2021 IOM
A positive integer is written on the board. Every minute Maxim adds to the number on the board one of its positive divisors, writes the result on the board and erases the previous number. However, it is forbidden for him to add the same number twice in a row. Prove that he can proceed in such a way that eventually a perfect square will appear on the board.
2021 IOM
Points $P$ and $Q$ are chosen on the side $BC$ of triangle $ABC$ so that $P$ lies between $B$ and $Q$. The rays $AP$ and $AQ$ divide the angle $BAC$ into three equal parts. It is known that the triangle $APQ$ is acute-angled. Denote by $B_1,P_1,Q_1,C_1$ the projections of points $B,P,Q,C$ onto the lines $AP,AQ,AP,AQ$, respectively. Prove that lines $B_1P_1$ and $C_1Q_1$ meet on line $BC$.
2021 IOM
Let $a_1,a_2,\ldots,a_n$ ($n\geq 2$) be nonnegative real numbers whose sum is $\frac{n}{2}$. For every $i=1,\ldots,n$ define

$$b_i=a_i+a_ia_{i+1}+a_ia_{i+1}a_{i+2}+\cdots+ a_ia_{i+1}\cdots a_{i+n-2}+2a_ia_{i+1}\cdots a_{i+n-1}$$where $a_{j+n}=a_j$ for every $j$. Prove that $b_i\geq 1$ holds for at least one index $i$.
2021 IOM
Six real numbers $x_1<x_2<x_3<x_4<x_5<x_6$ are given. For each triplet of distinct numbers of those six Vitya calculated their sum. It turned out that the $20$ sums are pairwise distinct; denote those sums by $$s_1<s_2<s_3<\cdots<s_{19}<s_{20}.$$It is known that $x_2+x_3+x_4=s_{11}$, $x_2+x_3+x_6=s_{15}$ and $x_1+x_2+x_6=s_{m}$. Find all possible values of $m$.
2021 IOM
There is a safe that can be opened by entering a secret code consisting of $n$ digits, each of them is $0$ or $1$. Initially, $n$ zeros were entered, and the safe is closed (so, all zeros is not the secret code).



In one attempt, you can enter an arbitrary sequence of $n$ digits, each of them is $0$ or $1$. If the entered sequence matches the secret code, the safe will open. If the entered sequence matches the secret code in more positions than the previously entered sequence, you will hear a click. In any other cases the safe will remain locked and there will be no click.



Find the smallest number of attempts that is sufficient to open the safe in all cases.
2021 IOM
Let $ABCD$ be a tetrahedron and suppose that $M$ is a point inside it such that $\angle MAD=\angle MBC$ and $\angle MDB=\angle MCA$. Prove that $$MA\cdot MB+MC\cdot MD<\max(AD\cdot BC,AC\cdot BD).$$
2020 IOM
In a triangle $ABC$ with a right angle at $C$, the angle bisector $AL$ (where $L$ is on segment $BC$) intersects the altitude $CH$ at point $K$. The bisector of angle $BCH$ intersects segment $AB$ at point $M$. Prove that $CK=ML$
2020 IOM
Does there exist a positive integer $n$ such that all its digits (in the decimal system) are greather than 5, while all the digits of $n^2$ are less than 5?
2020 IOM
Let $n>1$  be a given integer. The Mint issues coins of $n$ different values $a_1, a_2, ..., a_n$, where each $a_i$ is a positive integer (the number of coins of each value is unlimited). A set of values $\{a_1, a_2,..., a_n\}$ is called lucky, if the sum $a_1+ a_2+...+ a_n$ can be collected in a unique way (namely, by taking one coin of each value).

(a) Prove that there exists a lucky set of values $\{a_1, a_2, ..., a_n\}$ with $$a_1+ a_2+...+ a_n < n \cdot 2^n.$$(b) Prove that every lucky set of values $\{a_1, a_2,..., a_n\}$ satisfies $$a_1+ a_2+...+ a_n >n \cdot 2^{n-1}.$$

Proposed by Ilya Bogdanov
2020 IOM
Given three positive real numbers $a,b,c$ such that following holds $a^2=b^2+bc$, $b^2=c^2+ac$

Prove that $\frac{1}{c}=\frac{1}{a}+\frac{1}{b}$.
2020 IOM
There is an empty table with $2^{100}$ rows and $100$ columns. Alice and Eva take turns filling the empty cells of the first row of the table, Alice plays first. In each move, Alice chooses an empty cell and puts a cross in it; Eva in each move chooses an empty cell and puts a zero. When no empty cells remain in the first row, the players move on to the second row, and so on (in each new row Alice plays first).

The game ends when all the rows are filled. Alice wants to make as many different rows in the table as possible, while Eva wants to make as few as possible. How many different rows will be there in the table if both follow their best strategies?



Proposed by Denis Afrizonov
2020 IOM
In convex pentagon $ABCDE$ points $A_1$, $B_1$, $C_1$, $D_1$, $E_1$ are intersections of pairs of diagonals $(BD, CE)$, $(CE, DA)$, $(DA, EB)$, $(EB, AC)$ and $(AC, BD)$ respectively. Prove that if four of quadrilaterals $AB_{1}A_{1}B$, $BC_{1}B_{1}C$, $CD_{1}C_{1}D$, $DE_{1}D_{1}E$ and $EA_{1}E_{1}A$ are cyclic then the fifth one is also cyclic.
2019 IOM
Three prime numbers $p,q,r$ and a positive integer $n$ are given such that the numbers

$$ \frac{p+n}{qr}, \frac{q+n}{rp}, \frac{r+n}{pq} $$are integers. Prove that $p=q=r $.



Nazar Agakhanov
2019 IOM
In a social network with a fixed finite setback of users, each user had a fixed set of followers among the other users. Each user has an initial positive integer rating (not necessarily the same for all users). Every midnight, the rating of every user increases by the sum of the ratings that his followers had just before midnight.



Let $m$ be a positive integer. A hacker, who is not a user of the social network, wants all the users to have ratings divisible by $m$. Every day, he can either choose a user and increase his rating by 1, or do nothing. Prove that the hacker can achieve his goal after some number of days.



Vladislav Novikov
2019 IOM
In a non-equilateral triangle $ABC$ point $I$ is the incenter and point $O$ is the circumcenter. A line $s$ through $I$ is perpendicular to $IO$. Line $\ell$ symmetric to like $BC$ with respect to $s$ meets the segments $AB$ and $AC$ at points $K$ and $L$, respectively ($K$ and $L$ are different from $A$). Prove that the circumcenter of triangle $AKL$ lies on the line $IO$.



Dušan Djukić
2019 IOM
There are 100 students taking an exam. The professor calls them one by one and asks each student a single person question: “How many of 100 students will have a “passed” mark by the end of this exam?” The students answer must be an integer. Upon receiving the answer, the professor immediately publicly announces the student’s mark which is either “passed” or “failed.”



After all the students have got their marks, an inspector comes and checks if there is any student who gave the correct answer but got a “failed” mark. If at least one such student exists, then the professor is suspended and all the marks are replaced with “passed.” Otherwise no changes are made.



Can the students come up with a strategy that guarantees a “passed” mark to each of them?



 Denis Afrizonov
2019 IOM
We are given a convex four-sided pyramid with apex $S$ and base face $ABCD$ such that the pyramid has an inscribed sphere (i.e., it contains a sphere which is tangent to each race). By making cuts along the edges $SA,SB,SC,SD$ and rotating the faces $SAB,SBC,SCD,SDA$ outwards into the plane $ABCD$, we unfold the pyramid into the polygon $AKBLCMDN$ as shown in the figure. Prove that $K,L,M,N$ are concyclic.



 Tibor Bakos and Géza Kós
2019 IOM
Let $p$ be a prime and let $f(x)$ be a polynomial of degree $d$ with integer coefficients. Assume that the numbers $f(1),f(2),\dots,f(p)$ leave exactly $k$ distinct remainders when divided by $p$, and $1<k<p$. Prove that

$$ \frac{p-1}{d}\leq k-1\leq (p-1)\left(1-\frac1d \right) .$$

 Dániel Domán, Gauls Károlyi, and Emil Kiss
2018 IOM
Solve the system of equations in real numbers:

$$

\begin{cases*}

(x - 1)(y - 1)(z - 1) = xyz - 1,\\
(x - 2)(y - 2)(z - 2) = xyz - 2.\\
\end{cases*}

$$Vladimir Bragin
2018 IOM
A convex quadrilateral $ABCD$ is circumscribed about a circle $\omega$. Let $PQ$ be the diameter of $\omega$ perpendicular to $AC$. Suppose lines $BP$ and $DQ$ intersect at point $X$, and lines $BQ$ and $DP$ intersect at point $Y$. Show that the points $X$ and $Y$ lie on the line $AC$.



Géza Kós
2018 IOM
Let $k$ be a positive integer such that $p = 8k + 5$ is a prime number. The integers $r_1, r_2, \dots, r_{2k+1}$ are chosen so that the numbers $0, r_1^4, r_2^4, \dots, r_{2k+1}^4$ give pairwise different remainders modulo $p$. Prove that the product

$$\prod_{1 \leqslant i < j \leqslant 2k+1} \big(r_i^4 + r_j^4\big)$$is congruent to $(-1)^{k(k+1)/2}$ modulo $p$.



(Two integers are congruent modulo $p$ if $p$ divides their difference.)



Fedor Petrov
2018 IOM
Let $1 = d_0 < d_1 < \dots < d_m = 4k$ be all positive divisors of $4k$, where $k$ is a positive integer. Prove that there exists $i \in \{1, \dots, m\}$ such that $d_i - d_{i-1} = 2$.



Ivan Mitrofanov
2018 IOM
Ann and Max play a game on a $100 \times 100$ board.



First, Ann writes an integer from 1 to 10 000 in each square of the board so that each number is used exactly once.



Then Max chooses a square in the leftmost column and places a token on this square. He makes a number of moves in order to reach the rightmost column. In each move the token is moved to a square adjacent by side or vertex. For each visited square (including the starting one) Max pays Ann the number of coins equal to the number written in that square.



Max wants to pay as little as possible, whereas Ann wants to write the numbers in such a way to maximise the amount she will receive. How much money will Max pay Ann if both players follow their best strategies?



Lev Shabanov
2018 IOM
The incircle of a triangle $ABC$ touches the sides $BC$ and $AC$ at points $D$ and $E$, respectively. Suppose $P$ is the point on the shorter arc $DE$ of the incircle such that $\angle APE = \angle DPB$. The segments $AP$ and $BP$ meet the segment $DE$ at points $K$ and $L$, respectively. Prove that $2KL = DE$.



Dušan Djukić
2017 IOM
Let $ABCD$ be a parallelogram in which angle at $B$ is obtuse and $AD>AB$. Points $K$ and $L$ on $AC$ such that $\angle ADL=\angle KBA$(the points $A, K, C, L$ are all different, with $K$ between $A$ and $L$). The line $BK$ intersects the circumcircle $\omega$ of $ABC$ at points $B$ and $E$, and the line $EL$ intersects $\omega$ at points $E$ and $F$. Prove that $BF||AC$.
2017 IOM
In a country there are two-way non-stopflights between some pairs of cities. Any city can be reached from any other by a sequence of at most $100$ flights. Moreover, any city can be reached from any other by a sequence of an even number of flights. What is the smallest $d$ for which one can always claim that any city can be reached from any other by a sequence of an even number of flights not exceeding $d$?
2017 IOM
Let $Q$ be a quadriatic polynomial having two different real zeros. Prove that there is a non-constant monic polynomial $P$ such that all coefficients of the polynomial $Q(P(x))$ except the leading one are (by absolute value) less than $0.001$.
2017 IOM
Find the largest positive integer $N $ for which one can choose $N $ distinct numbers from the set ${1,2,3,...,100}$ such that neither the sum nor the product of any two different chosen numbers is divisible by $100$.



Proposed by Mikhail Evdokimov
2017 IOM
Let $x $ and $y $ be positive integers such that

$[x+2,y+2]-[x+1,y+1]=[x+1,y+1]-[x,y]$.Prove that one of the two numbers $x $ and $y $ divide the other.



(Here $[a,b] $ denote the least common multiple of $a $ and $b $).



Proposed by Dusan Djukic.
2017 IOM
et $ABCDEF$ be a convex hexagon which has an inscribed circle and a circumcribed. Denote by $\omega_{A}, \omega_{B},\omega_{C},\omega_{D},\omega_{E}$ and $\omega_{F}$ the inscribed circles of the triangles $FAB, ABC, BCD, CDE, DEF$ and $EFA$, respecitively. Let $l_{AB}$, be the external of $\omega_{A}$ and $\omega_{B}$; lines $l_{BC}$, $l_{CD}$, $l_{DE}$, $l_{EF}$, $l_{FA}$ are analoguosly defined. Let $A_1$ be the intersection point of the lines $l_{FA}$ and $l_{AB}$, $B_1, C_1, D_1, E_1, F_1$ are analogously defined.

Prove that $A_1D_1, B_1E_1, C_1F_1$ are concurrent.
2016 IOM
Find all positive integers $n$ such that there exist $n$ consecutive positive integers whose sum is a perfect square.