geometry problems from Flanders Junior Math Olympiads (JWO - Junior Wiskunde Olympiade)
with aops links
Two congruent right-angled isosceles triangles (with baselength 1) slide on a line as on the picture. What is the maximal area of overlap?
Through an internal point $O$ of $\Delta ABC$ one draws 3 lines, parallel to each of the sides, intersecting in the points shown on the picture.
Find the value of $\frac{|AF|}{|AB|}+\frac{|BE|}{|BC|}+\frac{|CN|}{|CA|}$.
Two $5\times1$ rectangles have 2 vertices in common as on the picture.
(a) Determine the area of overlap
(b) Determine the length of the segment between the other 2 points of intersection, $A$ and $B$.
Starting with two points A and B, some circles and points are constructed as shown in the figure:
the circle with centre A through B, the circle with centre B through A, the circle with centre C through A, the circle with centre D through B, the circle with centre E through A, the circle with centre F through A, the circle with centre G through A. Show that $M$ is the midpoint of $AB$.
2005 Flanders Juniors p4 (IMO Shortlist 1999 p1)
(a) Be M an internal point of the convex quadrilateral ABCD. Prove that $|MA|+|MB| < |AD|+|DC|+|CB|$.
(b) Be M an internal point of the triangle ABC. Note $k=\min(|MA|,|MB|,|MC|)$. Prove $k+|MA|+|MB|+|MC|<|AB|+|BC|+|CA|$.
A rectangular top measuring $12$ cm by $16$ cm becomes diagonal folded so that point $B$ coincides with point $A$. The line segment $[MN]$ shows the fold. Determine all possible positions of a point $P$ on the edge of the sheet such that the area of triangle $MNP$ is equal to $54$ cm$^2$.
In a triangle $ABC$ (with angles $\angle A,\angle B,\angle C$ and opposite sides $a, b, c$ respectively) we have $a = 4, b = 5$ and $\angle C = 2\angle A$. Determine $c$.
Consider the squares $ABCD, CEDF$ and $BGHE$ as in the figure on the right. Prove that $| AC | = | CH |$.
Consider a kite $ABCD$ with $| AB | = | AD | <| CB | = | CD |$. On $[BC]$ we choose $K$ so that $AK \parallel CD$. The intersection of $BD$ and $AK$ we call $X$. On $[CD]$ we choose $L$ such that $XL \parallel BC$. Prove that $\angle ABC = \angle AKL$.
The centers are connected in a square with side $1$ from the sides with the opposite vertices as in the figure. How big is the area of the octagon that arises?
Two circles with the same center have radii $r$ and $2r$. A chord $[AB]$ of the large circle intersects the small one at points $C$ and $D$ such that $| AC | = | CD | = | DB |$. Calculate length $| AB |$ in terms of $r$.
In a triangle $ABC$, $\angle B= 2\angle A$. Let $a = | BC |$, $b = | CA |$ and $c = | AB |$. Prove that $ac = b^2 - a^2$.
In the next figure a regular hexagon is given and $L, M$ and $N$ are the midpoints of $[AB], [CD]$ and $[EF]$, respectively. Show the area of the shaded area is equal to the area of the gray area.
Prove that there exists a positive number $\lambda$ such that for every triangle $ABC$, of which the medians from the vertices $B$ and $C$ intersect at right angle, holds $| AB |^2 + | AC |^2 = \lambda | BC |^2$.
The endpoints and the midpoints of two line segments are connected as shown in the figure.
Prove that the area of area $I$ is equal to the sum of the areas of areas $II$ and $III$.
Two lines $\ell$ and $m$ intersect perpendicularly at $O$. Along one side of $\ell$ take we take a point $F$ on $m$. Along the other side of $\ell$ we take a point $V$ that does not lie on $m$ .The line connecting the perpendicular projection of $V$ on $\ell$ with $F$ intersects the line $VO$ at $B$. Denote the distances from $F, V$ and $B$ to $\ell$ respectively $f, v$ and $b$. If we know that $f <v$, then prove that $$\frac{1}{f}=\frac{1}{v}+\frac{1}{b}$$
The points $M$ and $N$ lie respectively on the sides $[BC]$ and $[CD]$ of the square $ABCD$ such that $| CM | = | DN |$. The line segments $[DM]$ and $[BN]$ intersect each other in $P$. Prove that $AP \perp MN$.
On the side $[AC]$ of triangle $ABC$ lies a point $D$ with the following properties:
$\bullet$ $| BC | = | BD | $
$\bullet$ $\angle ABD = 2 \angle BAD$
$\bullet$ $BC$ is perpendicular to the bisector of $\angle ABD $.
Calculate the angles of $\vartriangle ABC$.
From a sheet of paper with length $a$ and width $b$ such that $b <a<2b$ is cut along a straight line a square $V_1$ that is as large as possible. Cut from the remaining rectangle a square $V_2$ that is as large as possible is again removed along a straight line.
(a) Prove that the perimeter of the now remaining rectangle is greater than one third of the perimeter of the original rectangle.
(b) Prove that the area of the remaining rectangle is no greater than $(3 - 2\sqrt2)$ times this of the original rectangle
In $\vartriangle ABC$, $| AB | = 6$ and $| AC | = 7$. Let $H$ be the orthogonal projection of $B$ on the bisector of $\angle A$ and call $M$ the midpoint of $[BC]$. Determine $| HM |$.
An acute-angled isosceles triangle $\angle ABC$ has apex $B$. A point $P$ lies on $[BC]$ such that $\angle CAP = 45^o$. The perpendicular bisector of $[AP]$ intersects the side $[AB]$ in $Q$. Prove that $PQ$ is perpendicular to $BC$.
Three line segments divide a triangle into five triangles.
The area of these triangles is called $u, v, x,$ yand $z$, as in the figure.
(a) Prove that $uv = yz$.
(b) Prove that the area of the great triangle is at most $ \frac{xz}{y}$
Given is $\vartriangle ABC$ (as in the figure) with semiperimeter $s$ and of which the lengths of the sides versus $A, B$ and $C$ are $a, b$ and $c$ respectively. At $[BC]$ we choose $K$ such that $| BK | = s - b$, on $[CA]$ we choose $L$ such that $| CL | = s - c$ and select $[AB]$ we $M$ such that $| AM | = s - a$. Find the angles of $\vartriangle KLM$ in terms of the angles $\alpha, \beta$ and $\gamma$ of the given triangle.
A circle with radius $1$ and a circle with radius $r> 1$ have the same center. The area of the small circle we call $A$. The area of the ring bounded by both circles we call $B$.
The triangle $\vartriangle UV W$ is right. A straight line cuts the hypotenuse $[V W]$ perpendicularly $X$ and cuts $[UW]$ into $Y$. Take $s> 1$ such that $| UV | = s \cdot | XY |$. The area of $\vartriangle XY W$. we call $C$ and the area of the quadrilateral $UV XY$ we call $D$.
Prove that $\sqrt{AC}+\sqrt{BD}=\sqrt{(A+C)(B+D)}$ if and only if $r = s$.
How many non-congruent pentagons exist of which $4$ angles are equal to $120^o$ and whose side lengths are five consecutive natural numbers (no necessarily ordered) ?
In the isosceles triangle $\vartriangle ABC$ with vertex angle $\angle B, F$ is a point on the side $[BC]$ such that $AF$ is the bisector of the angle of base $\angle A$. Also $| AF | + | FB | = | AC |$. Determine $\angle B$ .
In the quadrilateral $ABCD, E$ is the midpoint of $[AB]$. We know that $| CD | = 3, | BC | = 4, | BD | = 5$ and $| CE | = 6$. Show that $| AD | \ge 7$.
A pyramid with apex $T$ and base triangle $\vartriangle ABC$ has three right angles in the top. Determine the area of the base if the areas of the side faces are equal are to $8, 9$ and $12$.
In triangle $\vartriangle ABC$, $| AB | <| AC |$, $m$ is the perpendicular bisector of $[BC]$ and $s$ is the bisector from $\angle A$. The lines $m$ and $s$ intersect at the point $S$. The point $P$ is the foot of the perpendicular from $S$ on $AB$. The point $Q$ is the base of the perpendicular from $S$ to $AC$. Prove that $| AB | = | AQ | - | QC |$
In a parallelogram $ABCD$, the bisector of obtuse angle $\angle A$ intersects the side $[BC]$ at the point $P$ and the line $CD$ at the point $Q$. The perpendicular bisectors of triangle $\vartriangle PCQ$ intersect at point $O$. Prove that $\angle OBC = \angle ODC$ .
Consider a square $V W XY$ and an equilateral triangle $XY Z$ with $Z$ outside the square. The midpoint of $[V Y ]$ we call M. The mmidpoint of $[W Z]$ we call $N$. Find the measure of $\angle NMY$.
No comments:
Post a Comment