Olympiad problems

Durer Math Competition CD 1st Round - geometry, 2011.C4

Given a grid rectangle of size $2010 \times 1340$. A grid point is called [i]fair [/i] if the $2$ axis-parallel lines passing through it from the upper left and lower right corners of the large rectangle cut out a rectangle of equal area (such a point is shown in the figure). How many fair grid points lie inside the rectangle? [img]https://cdn.artofproblemsolving.com/attachments/1/b/21d4fb47c94b774994ac1c3aae7690bb98c7ae.png[/img]

Durer Math Competition CD 1st Round - geometry, 2009.D4

Tags: geometry , lattice points , area , Durer

If all vertices of a triangle on the square grid are grid points, then the triangle is called a [i]lattice[/i] triangle. What is the area of the lattice triangle with (one) of the smallest area, if one grid has area $1$ square unit?

Durer Math Competition CD 1st Round - geometry, 2013.D3

Tags: geometry , ratio , areas , Durer

The area of the triangle $ABC$ shown in the figure is $1$ unit. Points $D$ and $E$ lie on sides $AC$ and $BC$ respectively, and also are its ''one third'' points closer to $C$. Let $F$ be that $AE$ and $G$ are the midpoints of segment $BD$. What is the area of the marked quadrilateral $ABGF$? [img]https://cdn.artofproblemsolving.com/attachments/4/e/305673f429c86bbc58a8d40272dd6c9a8f0ab2.png[/img]

Durer Math Competition CD 1st Round - geometry, 2018.D2

Tags: geometry , similar , isosceles , Durer

In an isosceles triangle, we drew one of the angle bisectors. At least one of the resulting two smaller ones triangles is similar to the original. What can be the leg of the original triangle if the length of its base is $1$ unit?

Durer Math Competition CD 1st Round - geometry, 2017.C+5

Tags: geometry , Durer , Heptagon

Is there a heptagon and a point $P$ inside it such that any vertex of the heptagon has its distance from $P$ equal to the length of the side opposite the vertex? [i]A side and a vertex are said to be opposite if the side is the fourth from the vertex page (in any direction).[/i]

Durer Math Competition CD 1st Round - geometry, 2017.C1

Tags: Durer , geometry , angles , Decagon

The vertices of Durer's favorite regular decagon in clockwise order: $D_1, D_2, D_3, . . . , D_{10}$. What is the angle between the diagonals $D_1D_3$ and $D_2D_5$?

Durer Math Competition CD 1st Round - geometry, 2016.C+3

Tags: geometry , square , geometric inequality , Durer

Let $ABCD$ be a square with unit sides. Which interior point $P$ will the expression $\sqrt2 \cdot AP + BP + CP$ have a minimum value, and what is this minimum?

Durer Math Competition CD 1st Round - geometry, 2009.C3

Tags: geometry , altitudes , construction , altitude , Durer

We know the lengths of the $3$ altitudes of a triangle. Construct the triangle.

Durer Math Competition CD 1st Round - geometry, 2012.C5

Tags: geometry , Durer , incenter , Centroid

In a triangle, the line between the center of the inscribed circle and the center of gravity is parallel to one of the sides. Prove that the sidelengths form an arithmetic sequence.

Durer Math Competition CD 1st Round - geometry, 2014.C4

Tags: geometry , areas , pentagon , Durer

$ABCDE$ is a convex pentagon with $AB = CD = EA = 1$, $\angle ABC = \angle DEA = 90^o$, and $BC + DE = 1$. What is the area of the pentagon?

Durer Math Competition CD 1st Round - geometry, 2011.D5

Tags: geometry , circumcircle , convex polygon , Durer

Is it true that in every convex polygon $3$ adjacent vertices can be selected such that their circumcirscribed circle can cover the entire polygon?

Durer Math Competition CD 1st Round - geometry, 2008.C3

Tags: geometry , Squares , square , Durer

Given the squares $ABCD$ and $DEFG$, whose only common point is $D$. Let the midpoints of segments $AG$, $GE$, $EC$, and $CA$ be $H, I, J$, and $K$ respectively . Prove that $HIJK$ is a square. [img]https://cdn.artofproblemsolving.com/attachments/f/d/c3313e5bbf581977a74ea2b114d14950e38605.png[/img]

Durer Math Competition CD 1st Round - geometry, 2014.D3

Tags: geometry , octagon , regular octagon , Durer

$ABCDEF GH$ is a regular octagon with $10$ units side . The circle with center $A$ and radius $AC$ intersects the circle with center $D$ and radius $CD$ at point $ I$, different from $C$. What is the length of the segment $IF$?

Durer Math Competition CD 1st Round - geometry, 2012.D2

Tags: geometry , areas , Durer

Durer drew a regular triangle and then poked at an interior point. He made perpendiculars from it sides and connected it to the vertices. In this way, $6$ small triangles were created, of which (moving clockwise) all the second one is painted gray, as shown in figure. Show that the sum of the gray areas is just half the area of the triangle. [img]https://cdn.artofproblemsolving.com/attachments/e/7/a84ad28b3cd45bd0ce455cee2446222fd3eac2.png[/img]

Durer Math Competition CD 1st Round - geometry, 2015.D4

Tags: geometry , circumcircle , altitude , Durer

The altitude of the acute triangle $ABC$ drawn from $A$ , intersects the side $BC$ at $A_1$ and the circumscribed circle at $A_2$ (different from $A$). Similarly, we get the points $B_1$, $B_2$, $C_1$, $C_2$. Prove that $$\frac{AA_2}{AA_1}+\frac{BB_2}{BB_1}+\frac{CC_2}{CC_1}= 4.$$

Durer Math Competition CD 1st Round - geometry, 2021.C3

Tags: geometry , areas , Durer

Csenge has a yellow and a red foil on her rectangular window which look beautiful in the morning light. Where the two foils overlap, they look orange. The window is $80$ cm tall, $120$ cm wide and its corners are denoted by $A, B, C$ and $D$ in the figure. The two foils are triangular and both have two of their vertices at the two bottom corners of the window, A and $B$. The third vertex of the yellow foil is $S$, the trisecting point of side $DC$ closer to $D$, whereas the third vertex of the red foil is $P$, which is one fourth on the way on segment $SC$, closer to $C$. The red region (i.e. triangle $BPE$) is of area $16$ dm$^2$. What is the total area of the regions not covered by foil? [img]https://cdn.artofproblemsolving.com/attachments/b/c/ea371aeafde6968506da6f3456e88fa0bddc6d.png[/img]

Durer Math Competition CD 1st Round - geometry, 2014.C2

Tags: geometry , semicircles , equal segments , Durer

Above the segments $AB$ and $BC$ we drew a semicircle at each. $F_1$ bisects $AB$ and $F_2$ bisects $BC$. Above the segments $AF_2$ and $F_1C$ we also drew a semicircle at each. Segments $P Q$ and $RS$ touch the corresponding semicircles as shown in the figure. Prove that $P Q \parallel RS$ and $|P Q| = 2 \cdot |RS|$. [img]https://cdn.artofproblemsolving.com/attachments/8/2/570e923b91e9e630e3880a014cc6df4dc33aa2.png[/img]

Durer Math Competition CD 1st Round - geometry, 2012.D3

Tags: geometry , 3D geometry , cube , parallel , Durer

Show that the planes $ACG$ and $BEH$ defined by the vertices of the cube shown in Figure are parallel. What is their distance if the edge length of the cube is $1$ meter? [img]https://cdn.artofproblemsolving.com/attachments/c/9/21585f6c462e4289161b4a29f8805c3f63ff3e.png[/img]

Durer Math Competition CD 1st Round - geometry, 2016.C1

Tags: geometry , Concyclic , Durer

Let $P$ be an arbitrary point of the side line $AB$ of the triangle $ABC$. Mark the perpendicular projection of $P$ on the side lines $AC$ and $BC$ as $A_1$ and $B_1$ respectively. Denote $C_1$ he foot of the alttiude starting from $C$. Prove that the points $A_1$, $B_1$, $C_1$, $C$ and $P$ lie on a circle.

Durer Math Competition CD 1st Round - geometry, 2019.C3

Tags: geometry , areas , Durer

The best parts of grandma’s $30$ cm $ \times 30$ cm square shaped pie are the edges. For this reason grandma’s three grandchildren would like to split the pie between each other so that everyone gets the same amount (of the area) of the pie, but also of the edges. Can they cut the pie into three connected pieces like that?

Durer Math Competition CD 1st Round - geometry, 2022.C4

Tags: geometry , rectangle , Durer

We inscribed in triangle $ABC$ the rectangle $DEFG$ such that $D$ and $E$ fall on side $AB$, $F$ on side $BC$, and $G$ on side $AC$. We know that $AF$ bisects angle $\angle BAC$, and that $\frac{AD}{DE} = \frac12$. What is the measure of angle $\angle CAB$?

Durer Math Competition CD 1st Round - geometry, 2010.C3

Tags: geometry , reflection , Durer

The sides of a pool table are $3$ and $4$ meters long.We push a ball with an angle of $45^o$ at the sides. Is it true that it returns to where it started no matter where we started it from?

Durer Math Competition CD 1st Round - geometry, 2018.C3

Tags: geometry , midpoint , Durer , isosceles

In the isosceles triangle $ABC$, $AB = AC$. Let $E$ be on side $AB$ such that $\angle ACE = \angle ECB = 18^o$, and let $D$ be the midpoint of side $CB$. If we know the length of $AD$ is $3$ units, what is the length of $CE$?

Durer Math Competition CD 1st Round - geometry, 2017.D+2

Tags: geometry , trapezoid , Durer , collinear

Let the trapezoids $A_iB_iC_iD_i$ ($i = 1, 2, 3$) be similar and have the same clockwise direction. Their angles at $A_i$ and $B_i$ are $60^o$ and the sides $A_1B_1$, $B_2C_2$ and $A_3D_3$ are parallel. The lines $B_iD_{i+1}$ and $C_iA_{i+1}$ intersect at the point $P_i$ (the indices are understood cyclically, i.e. $A_4 = A_1$ and $D_4 = D_1$). Prove that the points $P_1$, $P_2$ and $P_3$ lie on a line.

Durer Math Competition CD 1st Round - geometry, 2018.D+4

Tags: geometry , incircle , tangent , Durer

The center of the inscribed circle of triangle $ABC$ is $I$. Let $e$ be the perpendicular line on $CI$ passing through $I$. The line $e$ itnersects the side $AC$ at $A'$ and the side $BC$ at point $B'$. Let $A''$ be the symmetric point of $A$ wrt $A'$, $B''$ be the symmetric point of $B$ wrt $B'$. Prove that $A''B''$ is a line tangent to the incircle.