Olympiad problems

2007 Sharygin Geometry Olympiad, 19

Tags: geometry , minimum , geometric inequality , angle , cyclic quadrilateral , area

Into an angle $A$ of size $a$, a circle is inscribed tangent to its sides at points $B$ and $C$. A line tangent to this circle at a point M meets the segments $AB$ and $AC$ at points $P$ and $Q$ respectively. What is the minimum $a$ such that the inequality $S_{PAQ}<S_{BMC}$ is possible?

Novosibirsk Oral Geo Oly IX, 2023.1

Tags: geometry , area

In the triangle $ABC$ on the sides $AB$ and $AC$, points $D$ and E are chosen, respectively. Can the segments $CD$ and $BE$ divide $ABC$ into four parts of the same area? [img]https://cdn.artofproblemsolving.com/attachments/1/c/3bbadab162b22530f1b254e744ecd068dea65e.png[/img]

Ukrainian TYM Qualifying - geometry, IV.8

Tags: geometric inequality , hexagon , convex , diagonal , area , geometry

Prove that in an arbitrary convex hexagon there is a diagonal that cuts off from it a triangle whose area does not exceed $\frac16$ of the area of the hexagon. What are the properties of a convex hexagon, each diagonal of which is cut off from it is a triangle whose area is not less than $\frac16$ the area of the hexagon?

1986 Tournament Of Towns, (107) 1

Tags: equal area , area , geometry

Through vertices $A$ and $B$ of triangle $ABC$ are constructed two lines which divide the triangle into four regions (three triangles and one quadrilateral). It is known that three of them have equal area. Prove that one of these three regions is the quadrilateral . (G . Galperin , A . Savin, Moscow)

1985 All Soviet Union Mathematical Olympiad, 395

Tags: geometry , hexagon , area

Two perpendiculars are drawn from the midpoints of each side of the acute-angle triangle to two other sides. Those six segments make hexagon. Prove that the hexagon area is a half of the triangle area.

1987 All Soviet Union Mathematical Olympiad, 458

Tags: convex , pentagon , diagonal , area

The convex $n$-gon ($n\ge 5$) is cut along all its diagonals. Prove that there are at least a pair of parts with the different areas.

1997 Israel Grosman Mathematical Olympiad, 2

Tags: geometry , polygon , area , lattice

Is there a planar polygon whose vertices have integer coordinates and whose area is $1/2$, such that this polygon is (a) a triangle with at least two sides longer than $1000$? (b) a triangle whose sides are all longer than $1000$? (c) a quadrangle?

2009 Bosnia and Herzegovina Junior BMO TST, 1

Tags: altitude , geometry , area , number theory

Lengths of sides of triangle $ABC$ are positive integers, and smallest side is equal to $2$. Determine the area of triangle $P$ if $v_c = v_a + v_b$, where $v_a$, $v_b$ and $v_c$ are lengths of altitudes in triangle $ABC$ from vertices $A$, $B$ and $C$, respectively.

2016 India PRMO, 5

Tags: geometry , area

Consider a triangle $ABC$ with $AB = 13, BC = 14, CA = 15$. A line perpendicular to $BC$ divides the interior of $\vartriangle BC$ into two regions of equal area. Suppose that the aforesaid perpendicular cuts $BC$ at $D$, and cuts $\vartriangle ABC$ again at $E$. If $L$ is the length of the line segment $DE$, find $L^2$.

2009 Oral Moscow Geometry Olympiad, 2

Tags: geometry , trapezoid , parallelogram , area

Trapezium $ABCD$ and parallelogram $MBDK$ are located so that the sides of the parallelogram are parallel to the diagonals of the trapezoid (see fig.). Prove that the area of the gray part is equal to the sum of the areas of the black part. (Yu. Blinkov) [img]https://cdn.artofproblemsolving.com/attachments/b/9/cfff83b1b85aea16b603995d4f3d327256b60b.png[/img]

2008 Flanders Math Olympiad, 4

Tags: square , geometry , circles , area

A square with sides $1$ and four circles of radius $1$ considered each having a vertex of have the square as the center. Find area of the shaded part (see figure). [img]https://cdn.artofproblemsolving.com/attachments/b/6/6e28d94094d69bac13c2702853ac2c906a80a1.png[/img]

1997 Tournament Of Towns, (526) 3

Tags: area , geometry

The vertical diameter of a circle is moved a centimetres to the right, and the horizontal diameter of this circle is moved $b$ centimetres up. These two lines divide the circle into four pieces. Consider the sum of the areas of the largest and the smallest pieces, and the sum of the areas of the other two pieces. Find the difference between these two sums. (G Galperin, NB Vassiliev)

1996 May Olympiad, 1

Tags: rectangle , area , geometry

Let $ABCD$ be a rectangle. A line $r$ moves parallel to $AB$ and intersects diagonal $AC$ , forming two triangles opposite the vertex, inside the rectangle. Prove that the sum of the areas of these triangles is minimal when $r$ passes through the midpoint of segment $AD$ .

2016 May Olympiad, 4

Tags: midline , area , geometry

In a triangle $ABC$, let $D$ and $E$ be points of the sides $ BC$ and $AC$ respectively. Segments $AD$ and $BE$ intersect at $O$. Suppose that the line connecting midpoints of the triangle and parallel to $AB$, bisects the segment $DE$. Prove that the triangle $ABO$ and the quadrilateral $ODCE$ have equal areas.

1992 Czech And Slovak Olympiad IIIA, 2

Tags: inequalities , geometric inequalities , area , 3d geometry , tetrahedron , geometry

Let $S$ be the total area of a tetrahedron whose edges have lengths $a,b,c,d, e, f$ . Prove that $S \le \frac{\sqrt3}{6} (a^2 +b^2 +...+ f^2)$

2009 Belarus Team Selection Test, 1

Tags: midpoint , geometry , geometric inequality , area

Let $M,N$ be the midpoints of the sides $AD,BC$ respectively of the convex quadrilateral $ABCD$, $K=AN \cap BM$, $L=CM \cap DN$. Find the smallest possible $c\in R$ such that $S(MKNL)<c \cdot S(ABCD)$ for any convex quadrilateral $ABCD$. I. Voronovich

2017 India PRMO, 26

Tags: chord , parallel , area , circles , geometry

Let $AB$ and $CD$ be two parallel chords in a circle with radius $5$ such that the centre $O$ lies between these chords. Suppose $AB = 6, CD = 8$. Suppose further that the area of the part of the circle lying between the chords $AB$ and $CD$ is $(m\pi + n) / k$, where $m, n, k$ are positive integers with gcd$(m, n, k) = 1$. What is the value of $m + n + k$ ?

May Olympiad L1 - geometry, 2017.3

Tags: geometry , rhombus , area

Let $ABCD$ be a rhombus of sides $AB = BC = CD= DA = 13$. On the side $AB$ construct the rhombus $BAFE$ outside $ABCD$ and such that the side $AF$ is parallel to the diagonal $BD$ of $ABCD$. If the area of $BAFE$ is equal to $65$, calculate the area of $ABCD$.

2004 Bosnia and Herzegovina Junior BMO TST, 2

Tags: geometry , rectangle , area

A rectangle is divided into $9$ smaller rectangles. The area of four of them is $5, 3, 9$ and $2$, as in the picture below. (The picture is not at scale.) [img]https://cdn.artofproblemsolving.com/attachments/8/e/0ccd6f41073f776b62e9ef4522df1f1639ee31.png[/img] Determine the minimum area of the rectangle. Under what circumstances is it achieved?

1993 IMO Shortlist, 7

Tags: geometry , circumcircle , quadrilateral , area

Let $A$, $B$, $C$, $D$ be four points in the plane, with $C$ and $D$ on the same side of the line $AB$, such that $AC \cdot BD = AD \cdot BC$ and $\angle ADB = 90^{\circ}+\angle ACB$. Find the ratio \[\frac{AB \cdot CD}{AC \cdot BD}, \] and prove that the circumcircles of the triangles $ACD$ and $BCD$ are orthogonal. (Intersecting circles are said to be orthogonal if at either common point their tangents are perpendicuar. Thus, proving that the circumcircles of the triangles $ACD$ and $BCD$ are orthogonal is equivalent to proving that the tangents to the circumcircles of the triangles $ACD$ and $BCD$ at the point $C$ are perpendicular.)

2000 Swedish Mathematical Competition, 4

Tags: geometry , lattice , area , geometric inequalities , 3d geometry , combinatorial geometry

The vertices of a triangle are three-dimensional lattice points. Show that its area is at least $\frac12$.

2001 IMO Shortlist, 6

Tags: geometry , linear algebra , area , triangle

Let $ABC$ be a triangle and $P$ an exterior point in the plane of the triangle. Suppose the lines $AP$, $BP$, $CP$ meet the sides $BC$, $CA$, $AB$ (or extensions thereof) in $D$, $E$, $F$, respectively. Suppose further that the areas of triangles $PBD$, $PCE$, $PAF$ are all equal. Prove that each of these areas is equal to the area of triangle $ABC$ itself.

1979 All Soviet Union Mathematical Olympiad, 270

Tags: coordinates , combinatorial geometry , area

A grasshopper is hopping in the angle $x\ge 0, y\ge 0$ of the coordinate plane (that means that it cannot land in the point with negative coordinate). If it is in the point $(x,y)$, it can either jump to the point $(x+1,y-1)$, or to the point $(x-5,y+7)$. Draw a set of such an initial points $(x,y)$, that having started from there, a grasshopper cannot reach any point farther than $1000$ from the point $(0,0)$. Find its area.

Estonia Open Junior - geometry, 2003.1.4

Tags: circles , geometry , area

Mari and Juri ordered a round pizza. Juri cut the pizza into four pieces by two straight cuts, none of which passed through the centre point of the pizza. Mari can choose two pieces not aside of these four, and Juri gets the rest two pieces. Prove that if Mari chooses the piece that covers the centre point of the pizza, she will get more pizza than Juri.

2015 Iran Geometry Olympiad, 4

Tags: geometry , parallelogram , rectangle , area

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