IFYM 2010-19 X-XII (Bulgaria) (-13R1, -17R2)

geometry problems from International Festival of Young Mathematicians, Sozopol, Bulgaria  (IFYM), grades 10-12, with aops links in the names

2010 - 2019 

missing are 2013 Round 1, 2017 Round 2 

2010 IFYM Round 1 p2
Let $A_1A_2A_3A_4A_5A_6A_7A_8$ be a right octagon with center $O$ and $\lambda_1$,$\lambda_2$, $\lambda_3$, $\lambda_4$ be some rational numbers for which:
$\lambda_1  \overrightarrow{OA_1}+\lambda_2 \overrightarrow{OA_2}+\lambda_3 \overrightarrow{OA_3}+\lambda_4 \overrightarrow{OA_4} =\overrightarrow{o}$.
Prove that $\lambda_1=\lambda_2=\lambda_3=\lambda_4=0$.

2010 IFYM Round 1 p3

Let $ABC$ is a triangle, let $H$ is orthocenter of $\triangle ABC$, let $M$ is midpoint of $BC$. Let $(d)$ is a line perpendicular with $HM$ at point $H$. Let $(d)$ meet $AB, AC$ at $E, F$ respectively. Prove that $HE =HF$.

2010 IFYM Round 1 p4
Let $ABCD$ be a square with side 1. On the sides $BC$ and $CD$ are chosen points $P$ and $Q$ where $AP$ and $AQ$ intersect the diagonal $BD$ in points $M$ and $N$ respectively. If $DQ\neq BP$ and the line through $A$ and the intersection point of $MQ$ and $NP$ is perpendicular to $PQ$, prove that $\angle MAN=45^\circ$.

2010 IFYM Round 1 p7
Let $\Delta ABC$ be an isosceles triangle with base $AB$. Point $P\in AB$ is such that $AP=2PB$. Point $Q$ from the segment $CP$ is such that $\angle AQP=\angle ACB$. Prove that $\angle PQB=\frac{1}{2}\angle ACB$.

2010 IFYM Round 2 p3
Two circles are intersecting in points $P$ and $Q$. Construct two points $A$ and $B$ on these circles so that $P\in AB$ and the product $AP.PB$ is maximal.

2010 IFYM Round 2 p7
Let $M$ be a convex polygon. Externally, on its sides are built squares. It is known that the vertices of these squares, that don’t lie on $M$, lie on a circle $k$. Determine $M$ (its type).

2010 IFYM Round 2 p8
In the trapezoid $ABCD, AB // CD$ and the diagonals intersect at $O$. The points $P, Q$ are on $AD, BC$ respectively such that $\angle AP B = \angle CP D$ and $\angle AQB = \angle CQD$. Show that $OP = OQ$.

2010 IFYM Round 3 p2
Let $ABCD$ be a quadrilateral, with an inscribed circle with center $I$. Through $A$ are constructed perpendiculars to $AB$ and $AD$, which intersect $BI$ and $DI$ in points $M$ and $N$ respectively. Prove that $MN\perp AC$.

2010 IFYM Round 3 p6
In $\Delta ABC$ $(AB>BC)$ $BM$ and $BL$ $(M,L\in AC)$ are a median and an angle bisector respectively. Let the line through $M$, parallel to $AB$, intersect $BL$ in point $D$ and the line through $L$, parallel to $BC$, intersect $BM$ in point $E$. Prove that $DE\perp BL$.

2010 IFYM Round 4 p1
The inscribed circle of $\Delta ABC$ is tangent to $AC$ and $BC$ in points $M$ and $N$ respectively. Line $MN$ intersects line $AB$ in point $P$, so that $B$ is between $A$ and $P$. Determine $\angle ABC$, if $BP=CM$.

2010 IFYM Round 4 p3
Through vertex $C$ of $\Delta ABC$ are constructed lines $l_1$ and $l_2$ which are symmetrical about the angle bisector $CL_c$. Prove that the projections of $A$ and $B$ on lines $l_1$ and $l_2$ lie on one circle.

2010 IFYM Round Final p5
We are given $\Delta ABC$, for which the excircle to side $BC$ is tangent to the continuations of $AB$ and $AC$ in points $E$ and $F$ respectively. Let $D$ be the reflection of $A$ in line $EF$. If it is known that $\angle BAC=2\angle BDC$, then determine $\angle BAC$.

2010 IFYM Round Final p8
Let $k$ be a circle and $l$–line that is tangent to $k$ in point $P$. On $l$ from the two sides of $P$ are chosen arbitrary points $A$ and $B$. The tangents through $A$ and $B$ to $k$, different than $l$, intersect in point $C$. Find the geometric place of points $C$, when $A$ and $B$ change in such way so that $AP.BP$ is a constant.

2011 IFYM Round 1 p2
Five distinct points $A,B,C,D$ and $E$ lie on a line with $|AB|=|BC|=|CD|=|DE|$. The point $F$ lies outside the line. Let $G$ be the circumcentre of the triangle $ADF$ and $H$ the circumcentre of the triangle $BEF$. Show that the lines $GH$ and $FC$ are perpendicular.

2011 IFYM Round 1 p5
A circle is inscribed in a quadrilateral $ABCD$, which is tangent to its sides $AB$, $BC$, $CD$, and $DC$ in points $M$, $N$, $P$, and $Q$ respectively. Prove that the lines $MP$, $NQ$, $AC$, and $BD$ intersect in one point.

2011 IFYM Round 1 p8
The lengths of the sides of a triangle are integers, whereas the radius of its circumscribed circle is a prime number. Prove that the triangle is right-angled.

2011 IFYM Round 2 p3
In a triangle $ABC$ a circle $k$ is inscribed, which is tangent to $BC$,$CA$,$AB$ in points $D,E,F$ respectively. Let point $P$ be inner for $k$. If the lines $DP$,$EP$,$FP$ intersect $k$ in points $D',E',F'$ respectively, then prove that $AD'$, $BE'$, and $CF'$ are concurrent.

2011 IFYM Round 2 p5
The vertices of $\Delta ABC$ lie on the graphics of the function $f(x)=x^2$ and its centroid is $M(1,7)$. Determine the greatest possible value of the area of $\Delta ABC$.

2011 IFYM Round 2 p7
The inscribed circle of $\Delta ABC$ $(AC<BC)$ is tangent to $AC$ and $BC$ in points $X$ and $Y$ respectively. A line is constructed through the middle point $M$ of $AB$, parallel to $XY$, which intersects $BC$ in $N$. Let $L\in BC$ be such that $NL=AC$ and $L$ is between $C$ and $N$. The lines $ML$ and $AC$ intersect in point $K$. Prove that $BN=CK$

2011 IFYM Round 3 p1
In triangle $ABC$ bisectors $AA_1$, $BB_1$ and $CC_1$ are drawn. Bisectors $AA_1$ and $CC_1$ intersect segments $C_1B_1$ and $B_1A_1$ at points $M$ and $N$, respectively. Prove that $\angle MBB_1 = \angle$$NBB_1$.

2011 IFYM Round 3 p3
Let $g_1$ and $g_2$ be some lines, which intersect in point $A$. A circle $k_1$ is tangent to $g_1$ at point $A$ and intersects $g_2$ for a second time in $C$. A circle $k_2$ is tangent to $g_2$ at point $A$ and intersects $g_1$ for a second time in $D$. The circles $k_1$ and $k_2$ intersect for a second time in point $B$. Prove that, if $\frac{AC}{AD}=\sqrt{2}$, then $\frac{BC}{BD}=2$.

2011 IFYM Round 4 p4
Let $A=\{P_1,P_2,…,P_{2011}\}$ be a set of points that lie in a circle $K(P_1,1)$. With $x_k$ we denote the distance between $P_k$ and the closest to it point from $A$. Prove that:
$\sum_{i=1}^{2011} x_i^2 \leq \frac{9}{4}$.

2011 IFYM Round Final p1
Let $ABCD$ be a quadrilateral inscribed in a circle $k$. Let the lines $AC\cap BD=O$, $AD\cap BC=P$, and $AB\cap CD=Q$. Line $QO$ intersects $k$ in points $M$ and $N$. Prove that $PM$ and $PN$ are tangent to $k$.

2011 IFYM Round Final p2
On side $AB$ of $\Delta ABC$ is chosen point $M$. A circle is tangent internally to the circumcircle of $\Delta ABC$ and segments $MB$ and $MC$ in points $P$ and $Q$ respectively. Prove that the center of the inscribed circle of $\Delta ABC$ lies on line $PQ$.


2012 IFYM Round 1 p2
In $\Delta ABC$ with $AC=10$ and $BC=15$ the points $G$ and $I$ are its centroid and the center of its inscribed circle respectively. Find $AB$, if $\angle GIC=90^\circ$.

2012 IFYM Round 1 p7
$\Delta ABC$ is such that $AC+BC=2$ and the sum of its altitude through $C$ and its base $AB$ is $CD+AB=\sqrt{5}$. Find the sides of the triangle.

2012 IFYM Round 2 p1
Find the area of a triangle with angles $\frac{1}{7} \pi$, $\frac{2}{7} \pi$, and $\frac{4}{7} \pi$, and radius of its circumscribed circle $R=1$.

2012 IFYM Round 2 p7
The quadrilateral $ABCD$ is such that $AB=AD=1$ and $\angle A=90^\circ$. If $CB=c$, $CA=b$, and $CD=a$, then prove that $(2-a^2-c^2 )^2+(2b^2-a^2-c^2 )^2=4a^2 c^2$ and $(a-c)^2\leq 2b^2\leq (a+c)^2$.

2012 IFYM Round 2 p8
The lengths of the sides of a convex decagon are no greater than 1. Prove that for each inner point $M$ of the decagon there is at least one vertex $A$, for which $MA\leq \frac{\sqrt{5}+1}{2}$.

2012 IFYM Round 3 p3
In a circle with radius 1 a regular n-gon $A_1 A_2...A_n$ is inscribed. Calculate the product $A_1 A_2.A_2 A_3A_n A_1$.

2012 IFYM Round 3 p7
Let $\Delta ABC$ be a triangle with orthocenter $H$ and midpoints $M_a,M_b$, and $M_c$ of $BC$, $AC$, and $AB$ respectively. A circle with center $H$ intersects the lines $M_bM_a$, $M_bM_c$, and $M_cM_a$ in points $U_1,U_2,V_1,V_2,W_1,W_2$ respectively. Prove that $CU_1=CU_2=AV_1=AV_2=BW_1=BW_2$.

2012 IFYM Round 3 p8
An equilateral triangle $ABC$ is inscribed in a square with side 1 (each vertex of the triangle is on a side of the square and no two are on the same side). Determine the greatest and smallest value of the side of $\Delta ABC$.

2012 IFYM Round 4 p4
In the right-angled $\Delta ABC$, with area $S$, a circle with area $S_1$ is inscribed and a circle with area $S_2$ is circumscribed. Prove the following inequality:
$\pi \frac{S-S_1}{S_2} <\frac{1}{\pi-1}$.

2012 IFYM Round 4 p6
Let $A_1 B_1 C_1$ and $A_2 B_2 C_2$ be two oppositely oriented concentric equilateral triangles. Prove that the lines $A_1 A_2$ , $B_1 B_2$ , and $C_1 C_2$ intersect in one point.

2012 IFYM Round 4 p7
A quadrilateral $ABCD$ is inscribed in a circle with center $O$. Let $A_1 B_1 C_1 D_1$ be the image of $ABCD$ after rotation with center $O$ and angle $\alpha \in (0,90^\circ)$. The points $P,Q,R$ and $S$ are intersections of $AB$ and $A_1 B_1$, $BC$ and $B_1 C_1$, $CD$ and $C_1 D_1$, and $DA$ and $D_1 A_1$. Prove that $PQRS$ is a parallelogram.

2012 IFYM Round Final p8
In a non-isosceles $\Delta ABC$ with angle bisectors $AL_a$, $BL_b$, and $CL_c$ we have that $L_aL_c=L_bL_c$. Prove that $\angle C$ is smaller than $120^\circ$.



2013 IFYM Round 1 p missing from aops


2013 IFYM Round 2 p2
Find the perimeter of the base of a regular triangular pyramid with volume $99$ and apothem $6$.

2013 IFYM Round 2 p7
Let $O$ be the center of the inscribed circle of $\Delta ABC$ and point $D$ be the middle point of $AB$. If $\angle AOD=90^\circ$, prove that $AB+BC=3AC$.

2013 IFYM Round 2 p8
Let $K$ be a point on the angle bisector, such that $\angle BKL=\angle KBL=30^\circ$. The lines $AB$ and $CK$ intersect in point $M$ and lines $AC$ and $BK$ intersect in point $N$. Determine $\angle AMN$.

2013 IFYM Round 3 p1
Point $D$ is from $AC$ of triangle $ABC$ so that $2AD=DC$. Let $DE$ be perpendicular to $BC$ and $AE$ intersects $BD$ at $F$. It is known that triangle $BEF$ is equilateral. Find $\angle ADB$.

2013 IFYM Round 4 p1
Let point $T$ be on side $AB$ of $\Delta ABC$ be such that $AT-BT=AC-BC$. The perpendicular from point $P$ to $AB$ intersects $AC$ in point $E$ and the angle bisectors of $\angle B$ and $\angle C$ intersect the circumscribed circle $k$ of $ABC$ in points $M$ and $L$. If $P$ is the second intersection point of the line $ME$ with $k$, then prove that $P,T,L$ are collinear.

2013 IFYM Round 4 p2
The point $P$, from the plane in which $\Delta ABC$ lies, is such that if $A_1,B_1$, and $C_1$ are the orthogonal projections of $P$ on the respective altitudes of $ABC$, then $AA_1=BB_1=CC_1=t$. Determine the locus of $P$ and length of $t$.

2013 IFYM Round Final p1
The points $P$ and $Q$ on the side $AC$ of the non-isosceles $\Delta ABC$ are such that
$\angle ABP=\angle QBC<\frac{1}{2}\angle ABC$. The angle bisectors of $\angle A$ and $\angle C$ intersect the segment $BP$ in points $K$ and $L$ and the segment $BQ$ in points $M$ and $N$, respectively. Prove that $AC$,$KN$, and $LM$ are concurrent.

2013 IFYM Round Final p2
Prove that for each $\Delta ABC$ with an acute $\angle C$ the following inequality is true:
$(a^2+b^2)  cos(\alpha -\beta )\leq 2ab$

2013 IFYM Round Final p8
Let $P$ be a polygon that is convex and symmetric to some point $O$. Prove that for some parallelogram $R$ satisfying $P\subset R$ we have $\frac{|R|}{|P|}\leq \sqrt 2$
where $|R|$ and $|P|$ denote the area of the sets $R$ and $P$, respectively.

by Witold Szczechla, Poland

2014 IFYM Round 1 p3
In an acute $\Delta ABC$, $AH_a$ and $BH_b$ are altitudes and $M$ is the middle point of $AB$. The circumscribed circles of $\Delta AMH_a$ and $\Delta BMH_b$ intersect for a second time in $P$. Prove that point $P$ lies on the circumscribed circle of $\Delta ABC$.

2014 IFYM Round 1 p7
If $AG_a,BG_b$, and $CG_c$ are symmedians in $\Delta ABC$ ($G_a\in BC,G_b\in AC,G_c\in AB$), is it possible for $\Delta G_a G_b G_c$ to be equilateral when $\Delta ABC$ is not equilateral?

2014 IFYM Round 2 p3
Nikolai and Peter are dividing a cake in the shape of a triangle. Firstly, Nikolai chooses one point $P$ inside the triangle and after that Peter cuts the cake by any line he chooses through $P$, then takes one of the pieces and leaves the other one for Nikolai. What’s the greatest portion of the cake Nikolai can be sure he could take, if he chooses $P$ in the best way possible?

2014 IFYM Round 2 p5
Let $ABCD$ be a convex quadrilateral. The rays $AB$ and $DC$ intersect in point $E$. Rays $AD$ and $BC$ intersect in point $F$. The angle bisector of $\angle DCF$ intersects $EF$ in point $K$. Let $I_1$ and $I_2$ be the centers of the inscribed circles in $\Delta ECB$ and $\Delta FCD$. $M$ is the projection of $I_2$ on line $CF$ and $N$ is the projection of $I_1$ on line $BC$. Let $P$ be the reflection of $N$ in  $I_1$. If $P,M,K$ are colinear, prove that $ABCD$ is tangential.

2014 IFYM Round 2 p7
In a convex quadrilateral $ABCD$, $\angle DAB=\angle BCD$ and the angle bisector of $\angle ABC$ passes through the middle point of $CD$. If $CD=3AD$, determine the ratio $\frac{AB}{BC}$.

2014 IFYM Round 3 p1
A line $l$ passes through the center $O$ of an equilateral triangle $\Delta ABC$, which intersects $CA$ in $N$ and $BC$ in $M$. Prove that we can construct a triangle with $AM$,$BN$, and $MN$ such that the altitude to $MN$ (in this triangle) is constant when $l$ changes.

2014 IFYM Round 3 p5
Let $\Delta ABC$ be an acute triangle. Points $P,Q\in AB$ so that $P$ is between $A$ and $Q$. Let $H_1$ and $H_2$ be the feet of the perpendiculars from $A$ to $CP$ and $CQ$ respectively. Let $H_3$ and $H_4$ be the feet of the perpendiculars from $B$ to $CP$ and $CQ$ respectively. Let $H_3 H_4\cap BC=X$ and $H_1 H_2\cap AC=Y$, so that $X$ is after $B$ and $Y$ is after $A$. If $XY\parallel AB$, prove that $CP$ and $CQ$ are isogonal to $\Delta ABC$.

2014 IFYM Round 4 p4
Let $\Delta ABC$ be a right triangle with $\angle ACB=90^\circ$. The points $P$ and $Q$ on the side $BC$ and $R$ and $S$ on the side $CA$ are such that $\angle BAP=\angle PAQ=\angle QAC$ and $\angle ABS=\angle SBR=\angle RBC$. If $AP\cap BS=T$, prove that $120^\circ<\angle RTB<150^\circ$.

2014 IFYM Round Final p5
Let $\Delta ABC$ be an acute triangle with $a>b$, center $O$ of its circumscribed circle and middle point $M$ of $AC$. Let $K$ be the reflection of $O$ in $M$. Point $E\in BC$ is such that $EO\perp AB$. Point $F\in MK$ is such that $FK=OE$ and $K$ lies between $F$ and $M$. The altitude through $C$ and the angle bisector of $\angle CAB$ intersect in $D$. Let $BD$ intersect the circumscribed circle of $\Delta ABC$ for a second time in $P$. Prove that $AP\perp CF$.

2015 IFYM Round 1 p6
The points $A_1$,$B_1$,$C_1$ are middle points of the arcs $\widehat{BC}, \widehat{CA}, \widehat{AB}$ of the circumscribed circle of $\Delta ABC$, respectively. The points $I_a,I_b,I_c$ are the reflections in the middle points of $BC,CA,AB$ of the center $I$ of the inscribed circle in the triangle. Prove that $I_a A_1,I_b B_1$, and $I_c C_1$ are concurrent.

2015 IFYM Round 1 p7
Let $ABCD$ be a trapezoid, where $AD\parallel BC$, $BC<AD$, and $AB\cap DC=T$. A circle $k_1$ is inscribed in $\Delta BCT$ and a circle $k_2$ is an excircle for $\Delta ADT$ which is tangent to $AD$ (opposite to $T$). Prove that the tangent line to $k_1$ through $D$, different than $DC$, is parallel to the tangent line to $k_2$ through $B$, different than $BA$.

2015 IFYM Round 2 p1
Let $AA_1$ be an altitude in $\Delta ABC$. Let $H_a$ be the orthocenter of the triangle with vertices the tangential points of the excircle to $\Delta ABC$, opposite to $A$. The points $B_1$, $C_1$, $H_b$, and $H_c$ are defined analogously. Prove that $A_1 H_a$, $B_1 H_b$, and $C_1 H_c$ are concurrent.

2015 IFYM Round 3 p1
Let ABCD be a convex quadrilateral such that $AB + CD = \sqrt{2}AC$ and $BC + DA = \sqrt{2}BD$. Prove that ABCD is a parallelogram.

2015 IFYM Round 3 p2
Let $ABCD$ be an inscribed quadrilateral and $P$ be an inner point for it so that $\angle PAB=\angle PBC=\angle PCD=\angle PDA$. The lines $AD$ and $BC$ intersect in point $Q$ and lines $AB$ and $CD$ – in point $R$. Prove that $\angle (PQ,PR)=\angle (AC,BD)$.

2015 IFYM Round 3 p8
The points $A_1,A_...,A_n$ lie on a circle with radius 1. The points $B_1,B_2,…,B_n$ are such that $B_i B_j<A_i A_j$ for $i\neq j$. Is it always true that the points $B_1,B_2,...,B_n$ lie on a circle with radius lesser than 1?

2015 IFYM Round 4 p3
The angle of a rotation $\rho$ is $\alpha <180^\circ$ and $\rho$ maps the convex polygon $M$ in itself. Prove that there exist two circles $c_1$ and $c_2$ with radius $r$ and $2r$, so that $c_1$ is inner for $M$ and $M$ is inner for $c_2$.

2015 IFYM Round 4 p7
In a square with side 1 are placed $n$ equilateral triangles (without having any parts outside the square) each with side greater than $\sqrt{\frac{2}{3}}$. Prove that all of the $n$ equilateral triangles have a common inner point.

2015 IFYM Round 4 p8
The quadrilateral $ABCD$ is circumscribed around a circle $k$ with center $I$ and $DA\cap CB=E$, $AB\cap DC=F$. In $\Delta EAF$ and $\Delta ECF$ are inscribed circles $k_1 (I_1,r_1)$ and $k_2 (I_2,r_2)$ respectively. Prove that the middle point $M$ of $AC$ lies on the radical axis of $k_1$ and $k_2$.

2015 IFYM Round Final p6
In $\Delta ABC$ points $A_1$, $B_1$, and $C_1$ are the tangential points of the excircles of $ABC$ with its sides.
a) Prove that $AA_1$, $BB_1$, and $CC_1$ intersect in one point $N$.
b) If $AC+BC=3AB$, prove that the center of the inscribed circle of $ABC$, its tangential point with $AB$, and the point $N$ are collinear.

2016 IFYM in aops (incomplete) 

[missing problems from Round 3, 4 finals]

2016 IFYM Round 1 p4

Let $ABCD$ be a convex quadrilateral. The circle $\omega_1$ is tangent to $AB$ in $S$ and the continuations after $A$ and $B$ of sides $DA$ and $CB$, circle $\omega_2$ with center $I$ is tangent to $BC$ and the continuations after $B$ and $C$ of sides $AB$ and $DC$, circle $\omega_3$ is tangent to $CD$ in $T$ and the continuations after $C$ and $D$ of sides $BC$ and $AD$, and circle $\omega_4$ with center $J$ is tangent to $DA$ and the continuations after $D$ and $A$ of sides $CD$ and $BA$. Prove that points $S$ and $T$ are on equal distance from the middle point of segment $IJ$.

2016 IFYM Round 2 p4

Circle $k$ passes through $A$ and intersects the sides of $\Delta ABC$ in $P,Q$, and $L$. Prove that:

$\frac{S_{PQL}}{S_{ABC}}\leq \frac{1}{4} (\frac{PL}{AQ})^2$.

2016 IFYM Round 2 p5

Points $K$ and $L$ are inner for $AB$ for an acute $\Delta ABC$, where $K$ is between $A$ and $L$. Let $P,Q$, and $H$ be the feet of the perpendiculars from $A$ to $CK$, from $B$ to $CL$, and from $C$ to $AB$, respectively. Point $M$ is the middle point of $AB$. If $PH\cap AC=X$ and $QH\cap BC=Y$, prove that points $H,P,M$, and $Q$ lie on one circle, if and only if the lines $AY,BX$, and $CH$ intersect in one point.

2016 IFYM Round 3 p3

Let $A_1 A_2…A_{66}$ be a convex 66-gon. What’s the greatest number of pentagons $A_i A_{i+1} A_{i+2} A_{i+3} A_{i+4},1\leq i\leq 66,$ which have an inscribed circle? ($A_{66+i}\equiv A_i$).

2016 IFYM Round 3 p5

We are given a $\Delta ABC$ with $\angle BAC=39^\circ$ and $\angle ABC=77^\circ$. Points $M$ and $N$ are chosen on $BC$ and $CA$ respectively, so that $\angle MAB=34^\circ$ and $\angle NBA=26^\circ$. Find $\angle BNM$.

2016 IFYM Round 3 p7

We are given a ruler with two marks at a distance 1. With its help we can do all possible constructions as with a ruler with no measurements, including one more: If there is a line $l$ and point $A$ on $l$, then we can construct points $P_1,P_2\in l$ for which $AP_1=AP_2=1$. By using this ruler, construct a perpendicular from a given point to a given line.

2016 IFYM Round 4 p3

The angle of a rotation $\rho$ is $\alpha <180^\circ$ and $\rho$ maps the convex polygon $M$ in itself. Prove that there exist two circles $c_1$ and $c_2$ with radius $r$ and $2r$, so that $c_1$ is inner for $M$ and $M$ is inner for $c_2$.

2016 IFYM Round 4 p5

A convex quadrilateral is cut into smaller convex quadrilaterals so that they are adjacent to each other only by whole sides.

a) Prove that if all small quadrilaterals are inscribed in a circle, then the original one is also inscribed in a circle.

b) Prove that if all small quadrilaterals are cyclic, then the original one is also cyclic

2016 IFYM Round Final p5

Prove that for an arbitrary $\Delta ABC$ the following inequality holds:

$\frac{l_a}{m_a}+\frac{l_b}{m_b}+\frac{l_c}{m_c} >1$, Where $l_a,l_b,l_c$ and $m_a,m_b,m_c$ are the lengths of the bisectors and medians through $A$, $B$, and $C$.

2016 IFYM Round Final p6

On the sides of a convex, non-regular $m$-gon are built externally regular heptagons. It is known that their centers are vertices of a regular $m$-gon. What’s the least possible value of $m$?

2016 IFYM Round Final p8 ?

2017 IFYM Round 1 p2
Point $F$ lies on the circumscribed circle around $\Delta ABC$, $P$ and $Q$ are projections of point $F$ on $AB$ and $AC$ respectively. Prove that, if $M$ and $N$ are the middle points of $BC$ and $PQ$ respectively, then $MN$ is perpendicular to $FN$.

2017 IFYM Round 1 p5
We are given a convex quadrilateral $ABCD$ with $AD=CD$ and $\angle BAD=\angle ABC.$
Points $K$ and $L$ are middle points of $AB$ and $BC$, respectively. The rays $\overrightarrow{DL}$ and $\overrightarrow{AB}$ intersect in $M$ and the rays $\overrightarrow{DK}$ and $\overrightarrow{BC}$ – in $N$. On segment $AN$ a point $X$ is chosen, such that $AX=CM$, and on segment $AC$ – point $Y$, such that $AY=MN$. Prove that the line $AB$ bisects segment $XY$.

2017 IFYM Round 2 missing from aops

2017 IFYM Round 3 p2
The lengths of the sides of a triangle are 19, 20, 21 cm. We can cut the triangle in a straight line into two parts. These two parts are put in a circle with radius $R$ cm without overlapping each other. Find the least possible value of $R$.

2017 IFYM Round 3 p7
The inscribed circle $\omega$ of an equilateral $\Delta ABC$ is tangent to its sides $AB$,$BC$ and $CA$ in points $D$,$E$, and $F$, respectively. Point $H$ is the foot of the altitude from $D$ to $EF$. Let $AH\cap BC=X,BH\cap CA=Y$. It is known that $XY\cap AB=T$. Let $D$ be the center of the circumscribed circle of $\Delta BYX$. Prove that $OH\perp CT$.

2017 IFYM Round 4 p1
$BB_1$ and $CC_1$ are altitudes in $\Delta ABC$. Let $B_1 C_1$ intersect the circumscribed circle of $\Delta ABC$ in points $E$ and $F$. Let $k$ be a circle passing through $E$ and $F$ in such way that the center of $k$ lies on the arc $\widehat{BAC}$. We denote with $M$ the middle point of $BC$. $X$ and $Y$ are the points on $k$ for which $MX$ and $MY$ are tangent to $k$. Let $EX\cap FY=S_1,EY\cap FX=S_2,BX\cap CY=U,$ and $BY\cap CX=V$. Prove that $S_1 S_2$ and $UV$ intersect in the orthocenter of $\Delta ABC$.

2017 IFYM Round 4 p7
Let $\Delta ABC$ be a scalene triangle with center $I$ of its inscribed circle. Points $A_1$,$B_1$, and $C_1$ are the points of tangency of the same circle with $BC$,$CA$, and $AB$ respectively. Prove that the circumscribed circles of $\Delta AIA_1$,$\Delta BIB_1$, and $\Delta CIC_1$ intersect in a common point, different from $I$.

2017 IFYM Round Final p3
$ABC$ is a triangle with a circumscribed circle $k$, center $I$ of its inscribed circle $\omega$, and center $I_a$ of its excircle $\omega _a$, opposite to $A$. $\omega$ and $\omega _a$ are tangent to $BC$ in points $P$ and $Q$, respectively, and $S$ is the middle point of the arc $\widehat{BC}$ that doesn’t contain $A$. Consider a circle that is tangent to $BC$ in point $P$ and to $k$ in point $R$. Let $RI$ intersect $k$ for a second time in point $L$. Prove that, $LI_a$ and $SQ$ intersect in a point that lies on $k$.

2017 IFYM Round Final p8
$k$ is the circumscribed circle of $\Delta ABC$. $M$ and $N$ are arbitrary points on sides $CA$ and $CB$, and $MN$ intersects $k$ in points $U$ and $V$. Prove that the middle points of $BM$,$AN$,$MN$, and $UV$ lie on one circle.

2018 IFYM Round 1 p7
For a non-isosceles $ABC$ we have that $2AC = AB + BC$. Point $I$ is the center of the circle inscribed in $\triangle ABC$, point $K$ is the middle of the arc $\widehat{AC}$ that includes point $B$, and point $T$ is from the line $AC$, such that $\angle TIB = 90^\circ$. Prove that the line $TB$ is tangent to the circumscribed circle of $\triangle KBI$.

2018 IFYM Round 2 p5
Point $X$ lies in a right-angled isosceles $\triangle ABC$ ($\angle ABC = 90^\circ$). Prove that $AX+BX+\sqrt{2}CX \geq \sqrt{5}AB$ and find for which points $X$ the equality is met.

2018 IFYM Round 2 p7
On the sides $AC$ and $AB$ of an acute $\triangle ABC$ are chosen points $M$ and $N$ respectively. Point $P$ is an intersection point of the segments $BM$ and $CN$ and point $Q$ is an inner point for the quadrilateral $ANPM$, for which $\angle BQC = 90^\circ$ and $\angle BQP = \angle BMQ$. If the quadrilateral $ANPM$ is inscribed in a circle, prove that $\angle QNC = \angle PQC$.

2018 IFYM Round 3 p3
The points $A$, $B$, $C$, $D$, and $E$ lie in one plane and have the following properties:
$AB = 12, BC = 50, CD = 38,  AD = 100, BE = 30, CE = 40$.
Find the length of the segment $ED$.

2018 IFYM Round 3 p5
On the sides $AB$,$BC$, and $CA$ of $\triangle ABC$ are chosen points $C_1$, $A_1$, and $B_1$ respectively, in such way that $AA_1$, $BB_1$, and $CC_1$ intersect in one point $X$. If $\angle A_1C_1B = \angle B_1C_1A$, prove that $CC_1$ is perpendicular to $AB$.

2018 IFYM Round 4 p1
In a quadrilateral $ABCD$ diagonal $AC$ is a bisector of $\angle BAD$ and $\angle ADC = \angle ACB$. The points $X$ and $Y$ are the feet of the perpendiculars from $A$ to $BC$ and $CD$ respectively. Prove that the orthocenter of $\triangle AXY$ lies on the line $BD$.

2018 IFYM Round Final p5
On the extension of the heights $AH_1$ and $BH_2$ of an acute $\triangle ABC$, after points $H_1$ and $H_2$, are chosen points $M$ and $N$ in such way that $\angle MCB = \angle NCA = 30^\circ$.
We denote with $C_1$ the intersection point of the lines $MB$ and $NA$. Analogously we define $A_1$ and $B_1$. Prove that the straight lines $AA_1$, $BB_1$, and $CC_1$ intersect in one point.


2019 IFYM Round 1 p3
The perpendicular bisector of $AB$ of an acute $\Delta ABC$ intersects $BC$ and the continuation of $AC$ in points $P$ and $Q$ respectively. $M$ and $N$ are the middle points of side $AB$ and segment $PQ$ respectively. If the lines $AB$ and $CN$ intersect in point $D$, prove that $\Delta ABC$ and $\Delta DCM$ have a common orthocenter.

2019 IFYM Round 1 p4
The diagonals $AC$ and $BD$ of a convex quadrilateral $ABCD$ intersect in point $M$. The angle bisector of $\angle ACD$ intersects the ray $\overrightarrow{BA}$ in point $K$. If
$MA.MC+MA.CD=MB.MD$, prove that $\angle BKC=\angle CDB$.

2019 IFYM Round 2 p2
In $\Delta ABC$ with $\angle ACB=135^\circ$, are chosen points $M$ and $N$ on side $AB$, so that $\angle MCN=90^\circ$. Segments $MD$ and $NQ$ are angle bisectors of $\Delta AMC$ and $\Delta NBC$ respectively. Prove that the reflection of $C$ in line $PQ$ lies on the line $AB$.

2019 IFYM Round 2 p4
For a quadrilateral $ABCD$ is given that $\angle CBD=2\angle ADB$, $\angle ABD=2\angle CDB$, and $AB=CB$. Prove that $AD=CD$.

2019 IFYM Round 3 p2
$\Delta ABC$ is a triangle with center $I$ of its inscribed circle and $B_1$ and $C_1$ are feet of its angle bisectors through $B$ and $C$. Let $S$ be the middle point on the arc $\widehat{BAC}$ of the circumscribed circle of $\Delta ABC$ (denoted with $\Omega$) and let $\omega_a$ be the excircle of $\Delta ABC$ opposite to $A$. Let $\omega_a (I_a)$ be tangent to $AB$ and $AC$ in points $D$ and $E$ respectively and $SI\cap \Omega=\{S,P\}$. Let $M$ be the middle point of $DE$ and $N$ be the middle point of $SI$. If $MN\cap AP=K$, prove that $KI_a\perp B_1 C_1$.

2019 IFYM Round 3 p3
We are given a non-obtuse $\Delta ABC$ $(BC>AC)$ with an altitude $CD$ $(D\in AB)$, center $O$ of its circumscribed circle, and a middle point $M$ of its side $AB$. Point $E$ lies on the ray $\overrightarrow{BA}$ in such way that $AE.BE=DE.ME$. If the line $OE$ bisects the area of $\Delta ABC$ and $CO=CD.cos\angle ACB$, determine the angles of $\Delta ABC$.

2019 IFYM Round 4 p1
The points $M$ and $N$ are on the side $BC$ of $\Delta ABC$, so that $BM=CN$ and $M$ is between $B$ and $N$. Points $P\in AN$ and $Q\in AM$ are such that $\angle PMC=\angle MAB$ and $\angle QNB=\angle NAC$. Prove that $\angle QBC=\angle PCB$.

2019 IFYM Round 4 p8
We are given a $\Delta ABC$. Point $D$ on the circumscribed circle k is such that $CD$ is a symmedian in $\Delta ABC$. Let $X$ and $Y$ be on the rays $\overrightarrow{CB}$ and $\overrightarrow{CA}$, so that $CX=2CA$ and $CY=2CB$. Prove that the circle, tangent externally to $k$ and to the lines $CA$ and $CB$, is tangent to the circumscribed circle of $\Delta XDY$.

2019 IFYM Round Final p3
$\Delta ABC$ is isosceles with a circumscribed circle $\omega (O)$. Let $H$ be the foot of the altitude from $C$ to $AB$ and let $M$ be the middle point of $AB$. We define a point $X$ as the second intersection point of the circle with diameter $CM$ and $\omega$ and let $XH$ intersect $\omega$ for a second time in $Y$. If $CO\cap AB=D$, then prove that the circumscribed circle of $\Delta YHD$ is tangent to $\omega$.

2019 IFYM Round Final p4
The inscribed circle of an acute $\Delta ABC$ is tangent to $AB$ and $AC$ in $K$ and $L$ respectively. The altitude $AH$ intersects the angle bisectors of $\angle ABC$ and $\angle ACB$ in $P$ and $Q$ respectively. Prove that the middle point $M$ of $AH$ lies on the radical axis of the circumscribed circles of $\Delta KPB$ and $\Delta LQC$.