Elementary

1. With putting the four shapes drawn in the following figure together make a shape with at least two reflection symmetries.
2. Points $K, L, M, N$ lie on the sides $AB, BC, CD, DA$ of a square $ABCD$, respectively, such that the area of $KLMN$ is equal to one half of the area of $ABCD$. Prove that some diagonal of $KLMN$ is parallel to some side of $ABCD$.
3. As shown in the following figure, a heart is a shape consist of three semicircles with diameters $AB$, $BC$ and $AC$ such that $B$ is midpoint of the segment $AC$. A heart $\omega$ is given. Call a pair $(P, P')$ bisector if $P$ and $P'$ lie on $\omega$ and bisect its perimeter. Let $(P, P')$ and $(Q,Q')$ be bisector pairs. Tangents at points $P, P', Q$, and $Q'$ to $\omega$ construct a convex quadrilateral $XYZT$. If the quadrilateral $XYZT$ is inscribed in a circle, find the angle between lines $PP'$ and $QQ'$.
4. In isosceles trapezoid $ABCD$ ($AB \parallel CD$) points $E$ and $F$ lie on the segment $CD$ in such a way that $D, E, F$ and $C$ are in that order and $DE = CF$. Let $X$ and $Y$ be the reflection of $E$ and $C$ with respect to $AD$ and $AF$. Prove that circumcircles of triangles $ADF$ and $BXY$ are concentric.
5. Let $A_1, A_2, . . . , A_{2021}$ be $2021$ points on the plane, no three collinear and$$\angle A_1A_2A_3 + \angle A_2A_3A_4 +... + \angle A_{2021}A_1A_2 = 360^o,$$in which by the angle $\angle A_{i-1}A_iA_{i+1}$ we mean the one which is less than $180^o$ (assume that $A_{2022} =A_1$ and $A_0 = A_{2021}$). Prove that some of these angles will add up to $90^\circ$.

Intermediate

1. Let $ABC$ be a triangle with $AB = AC$. Let $H$ be the orthocenter of $ABC$. Point $E$ is the midpoint of $AC$ and point $D$ lies on the side $BC$ such that $3CD = BC$. Prove that $BE \perp HD$.
2. Let $ABCD$ be a parallelogram. Points $E, F$ lie on the sides $AB, CD$ respectively, such that $\angle EDC = \angle FBC$ and $\angle ECD = \angle FAD$. Prove that $AB \geq 2BC$.
3. Given a convex quadrilateral $ABCD$ with $AB = BC$and $\angle ABD = \angle BCD = 90$.Let point $E$ be the intersection of diagonals $AC$ and $BD$. Point $F$ lies on the side $AD$ such that $\frac{AF}{F D}=\frac{CE}{EA}$.. Circle $\omega$ with diameter $DF$ and the circumcircle of triangle $ABF$ intersect for the second time at point $K$. Point $L$ is the second intersection of $EF$ and $\omega$. Prove that the line $KL$ passes through the midpoint of $CE$.
4. Let $ABC$ be a scalene acute-angled triangle with its incenter $I$ and circumcircle $\Gamma$. Line $AI$ intersects $\Gamma$ for the second time at $M$. Let $N$ be the midpoint of $BC$ and $T$ be the point on $\Gamma$ such that $IN \perp MT$. Finally, let $P$ and $Q$ be the intersection points of $TB$ and $TC$, respectively, with the line perpendicular to $AI$ at $I$. Show that $PB = CQ$.
5. Consider a convex pentagon $ABCDE$ and a variable point $X$ on its side $CD$. Suppose that points $K, L$ lie on the segment $AX$ such that $AB = BK$ and $AE = EL$ and that the circumcircles of triangles $CXK$ and $DXL$ intersect for the second time at $Y$ . As $X$ varies, prove that all such lines $XY$ pass through a fixed point, or they are all parallel.

1. Acute-angled triangle $ABC$ with circumcircle $\omega$ is given. Let $D$ be the midpoint of $AC$, $E$ be the foot of altitude from $A$ to $BC$, and $F$ be the intersection point of $AB$ and $DE$. Point $H$ lies on the arc $BC$ of $\omega$ (the one that does not contain $A$) such that $\angle BHE=\angle ABC$. Prove that $\angle BHF=90^\circ$.
2. Two circles $\Gamma_1$ and $\Gamma_2$ meet at two distinct points $A$ and $B$. A line passing through $A$ meets $\Gamma_1$ and $\Gamma_2$ again at $C$ and $D$ respectively, such that $A$ lies between $C$ and $D$. The tangent at $A$ to $\Gamma_2$ meets $\Gamma_1$ again at $E$. Let $F$ be a point on $\Gamma_2$ such that $F$ and $A$ lie on different sides of $BD$, and $2\angle AFC=\angle ABC$. Prove that the tangent at $F$ to $\Gamma_2$, and lines $BD$ and $CE$ are concurrent.
3. Consider a triangle $ABC$ with altitudes $AD, BE$, and $CF$, and orthocenter $H$. Let the perpendicular line from $H$ to $EF$ intersects $EF, AB$ and $AC$ at $P, T$ and $L$, respectively. Point $K$ lies on the side $BC$ such that $BD=KC$. Let $\omega$ be a circle that passes through $H$ and $P$, that is tangent to $AH$. Prove that circumcircle of triangle $ATL$ and $\omega$ are tangent, and $KH$ passes through the tangency point.
4. $2021$ points on the plane in the convex position, no three collinear and no four concyclic, are given. Prove that there exist two of them such that every circle passing through these two points contains at least $673$ of the other points in its interior. (A finite set of points on the plane are in convex position if the points are the vertices of a convex polygon.)
5. Given a triangle $ABC$ with incenter $I$. The incircle of triangle $ABC$ is tangent to $BC$ at $D$. Let $P$ and $Q$ be points on the side BC such that $\angle PAB = \angle BCA$ and $\angle QAC = \angle ABC$, respectively. Let $K$ and $L$ be the incenter of triangles $ABP$ and $ACQ$, respectively. Prove that $AD$ is the Euler line of triangle $IKL$.
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