Let $ABC$ be a triangle with $AC > BC,$ let $\omega$ be the circumcircle of $\triangle ABC,$ and let $r$ be its radius. Point $P$ is chosen on $\overline{AC}$ such taht $BC=CP,$ and point $S$ is the foot of the perpendicular from $P$ to $\overline{AB}$. Ray $BP$ mets $\omega$ again at $D$. Point $Q$ is chosen on line $SP$ such that $PQ = r$ and $S,P,Q$ lie on a line in that order. Finally, let $E$ be a point satisfying $\overline{AE} \perp \overline{CQ}$ and $\overline{BE} \perp \overline{DQ}$. Prove that $E$ lies on $\omega$.

Let $A, B$ and $C$ three points on a line $l$, in that order .Let $D$ a point outside $l$ and $\Gamma$ the circumcircle of $\triangle BCD$, the tangents from $A$ to $\Gamma$ touch $\Gamma$ on $S$ and $T$. Let $P$ the intersection of $ST$ and $AC$. Prove that $P$ does not depend of the choice of $D$.

A large rectangle is partitioned into four rectangles by two segments parallel to its sides. The areas of three of the resulting rectangles are shown. What is the area of the fourth rectangle? [asy] draw((0,0)--(10,0)--(10,7)--(0,7)--cycle); draw((0,5)--(10,5)); draw((3,0)--(3,7)); label("6", (1.5,6)); label("?", (1.5,2.5)); label("14", (6.5,6)); label("35", (6.5,2.5)); [/asy] $ \textbf{(A)}\ 10 \qquad\textbf{(B)}\ 15 \qquad\textbf{(C)}\ 20 \qquad\textbf{(D)}\ 21 \qquad\textbf{(E)}\ 25 $

A collection of cardboard circles, each with a diameter of at most $1$, lie on a $5\times 8$ table without overlapping or overhanging the edge of the table. A cardboard circle of diameter $2$ is added to the collection. Prove that this new collection of cardboard circles can be placed on a $7\times 7$ table without overlapping or overhanging the edge.

Let $ \,ABC\,$ be a triangle and $ \,P\,$ an interior point of $ \,ABC\,$. Show that at least one of the angles $ \,\angle PAB,\;\angle PBC,\;\angle PCA\,$ is less than or equal to $ 30^{\circ }$.

Consider a trapezium $AB \Gamma \Delta$, where $A\Delta \parallel B\Gamma$ and $\measuredangle A = 120^{\circ}$. Let $E$ be the midpoint of $AB$ and let $O_1$ and $O_2$ be the circumcenters of triangles $AE \Delta$ and $BE\Gamma$, respectively. Prove that the area of the trapezium is equal to six time the area of the triangle $O_1 E O_2$.

Let $ABC$ be a triangle, and let $D, E$ and $F$ be the feet of the altitudes from $A, B$ and $C,$ respectively. A circle $\omega_A$ through $B$ and $C$ crosses the line $EF$ at $X$ and $X'$. Similarly, a circle $\omega_B$ through $C$ and $A$ crosses the line $FD$ at $Y$ and $Y',$ and a circle $\omega_C$ through $A$ and $B$ crosses the line $DE$ at $Z$ and $Z'$. Prove that $X, Y$ and $Z$ are collinear if and only if $X', Y'$ and $Z'$ are collinear. [i]Vlad Robu[/i]

Prove that for any real $r>0$, one can cover the circumference of a $1\times r$ rectangle with non-intersecting disks of unit radius.

Let $[ABC]$ be a triangle and $D$ a point between $A$ and $B$. If the triangles $[ABC], [ACD]$ and $[BCD]$ are all isosceles, what are the possible values of $\angle ABC$?

On the sides $AB$, $BC$, $CD$, $DA$ of the square $ABCD$ points $P, Q, R, T$ are chosen such that $$\frac{AP}{PB}=\frac{BQ}{QC}=\frac{CR}{RD}=\frac{DT}{TA}=\frac12.$$ The segments $AR$, $BT$, $CP$, $DQ$ in the intersection form the quadrilateral $KLMN$ (see figure). [img]https://cdn.artofproblemsolving.com/attachments/f/c/587a2358734c300fe7082c520f90c91f872b49.png[/img] a) Prove that $KLMN$ is a square. b) Find the ratio of the areas of the squares $KLMN$ and $ABCD$. (Alexander Shkolny)

$ABC$ is an equilateral triangle with side length $1$. Point $D$ lies on $\overline{AB}$, point $E$ lies on $\overline{AC}$, and points $G$ and $F$ lie on $\overline{BC}$, such that $DEFG$ is a square. What is the area of $DEFG$?

Inside a $2\times 2$ square, lines parallel to a side of the square (both horizontal and vertical) are drawn thereby dividing the square into rectangles. The rectangles are alternately colored black and white like a chessboard. Prove that if the total area of the white rectangles is equal to the total area of the black rectangles, then one can cut out the black rectangles and reassemble them into a $1\times 2$ rectangle.

[b]p1.[/b] For $25$ bagels they paid as many rubles as the number of bagels you can buy with a ruble. How much does one bagel cost? [b]p2.[/b] Cut the square into the figure into$ 4$ parts of the same shape and size so that each part contains exactly one shaded square. [img]https://cdn.artofproblemsolving.com/attachments/a/2/14f0d435b063bcbc55d3dbdb0a24545af1defb.png[/img] [b]p3.[/b] The numerator and denominator of the fraction are positive numbers. The numerator is increased by $1$, and the denominator is increased by $10$. Can this increase the fraction? [b]p4.[/b] The brother left the house $5$ minutes later than his sister, following her, but walked one and a half times faster than her. How many minutes after leaving will the brother catch up with his sister? [b]p5.[/b] Three apples are worth more than five pears. Can five apples be cheaper than seven pears? Can seven apples be cheaper than thirteen pears? (All apples cost the same, all pears too.) [b]p6.[/b] Give an example of a natural number divisible by $6$ and having exactly $15$ different natural divisors (counting $1$ and the number itself). [b]p7.[/b] In a round dance, $30$ children stand in a circle. Every girl's right neighbor is a boy. Half of the boys have a boy on their right, and all the other boys have a girl on their right. How many boys and girls are there in a round dance? [b]p8.[/b] A sheet of paper was bent in half in a straight line and pierced with a needle in two places, and then unfolded and got $4$ holes. The positions of three of them are marked in figure Where might the fourth hole be? [img]https://cdn.artofproblemsolving.com/attachments/c/8/53b14ddbac4d588827291b27c40e3f59eabc24.png[/img] [b]p9 [/b] The numbers 1$, 2, 3, 4, 5, _, 2000$ are written in a row. First, third, fifth, etc. crossed out in order. Of the remaining $1000 $ numbers, the first, third, fifth, etc. are again crossed out. They do this until one number remains. What is this number? [b]p10.[/b] On the number axis there lives a grasshopper who can jump $1$ and $4$ to the right and left. Can he get from point $1$ to point $2$ of the numerical axis in $1996$ jumps if he must not get to points with coordinates divisible by $4$ (points $0$, $\pm 4$, $\pm 8$ etc.)? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics]here.[/url]

Let $ ABCD$ be a quadrilateral in which $ AB$ is parallel to $ CD$ and perpendicular to $ AD; AB \equal{} 3CD;$ and the area of the quadrilateral is $ 4$. if a circle can be drawn touching all the four sides of the quadrilateral, find its radius.

A tetrahedron $ABCD$ with acute-angled faces is inscribed in a sphere with center $O$. A line passing through $O$ perpendicular to plane $ABC$ crosses the sphere at point $D'$ that lies on the opposide side of plane $ABC$ than point $D$. Line $DD'$ crosses plane $ABC$ in point $P$ that lies inside the triangle $ABC$. Prove, that if $\angle APB=2\angle ACB$, then $\angle ADD'=\angle BDD'$.

Let $\vartriangle ABC$ be a triangle such that $\angle ABC = 30^o$, $\angle ACB = 15^o$. Let $M$ be midpoint of segment $BC$ and point $N$ lies on segment $MC$, such that the length of $NC$ is equal to length of $AB$. Proce that $AN$ is the bisector of the angle $\angle MAC$. [img]https://cdn.artofproblemsolving.com/attachments/2/7/4c554b53f03288ee69931fdd2c6fbd3e27ab13.png[/img]

Let $ABCDEF$ be a regular hexagon, and let $P$ be a point inside quadrilateral $ABCD$. If the area of triangle $PBC$ is $20$, and the area of triangle $PAD$ is $23$, compute the area of hexagon $ABCDEF$.

Let $a, b, c$ be the lengths of the sides of a triangle, and let $S$ be it's area. Prove that $$S \le \frac{a^2+b^2+c^2}{4\sqrt3}$$ and the equality is achieved only for an equilateral triangle.

A $ABC$ triangle divide by a $d$ line such that, new two pieces' areas are equal. $d$ line intersects with $[AB]$ at $D$, $[AC]$ at $E$. Prove that $$\frac{AD+AE}{BD+DE+EC+CB} > \frac{1}{4}$$

Given parallelogram $ABCD$, construct point $F$ so that $CF\perp BC$, as shown. Also $F$ is placed so that $\angle DFC = 120^o$. If $DF = 4$ and $BC =CF = 2$, what is the area of the parallelogram? [img]https://cdn.artofproblemsolving.com/attachments/7/4/0cdb0752760686acb891580da55f55212098fb.jpg[/img]

The figure below shows a smaller square within a larger square. Both squares have integer side lengths. The region inside the larger square but outside the smaller square has area $52$. Find the area of the larger square. [img]https://cdn.artofproblemsolving.com/attachments/a/f/2cb8c70109196bf30f88aef0c53bbac07d6cc3.png[/img]

We have six pairwise non-intersecting circles that the radius of each is at least one (no circle lies in the interior of any other circle). Prove that the radius of any circle intersecting all the six circles, is at least one. [i]Proposed by Mohammad Ali Abam - Morteza Saghafian[/i]

In the plane, there are $n \geqslant 6$ pairwise disjoint disks $D_{1}, D_{2}, \ldots, D_{n}$ with radii $R_{1} \geqslant R_{2} \geqslant \ldots \geqslant R_{n}$. For every $i=1,2, \ldots, n$, a point $P_{i}$ is chosen in disk $D_{i}$. Let $O$ be an arbitrary point in the plane. Prove that \[O P_{1}+O P_{2}+\ldots+O P_{n} \geqslant R_{6}+R_{7}+\ldots+R_{n}.\] (A disk is assumed to contain its boundary.)

Is there a convex pentagon in which each diagonal is equal to some side?

A rectangular sheet of paper (white on one side and gray on the other) was folded three times, as shown in the figure: Rectangle $1$, which was white after the first fold, has $20$ cm more perimeter than rectangle $2$, which was white after the second fold, and this in turn has $16$ cm more perimeter than rectangle $3$, which was white after the third fold. Determine the area of the sheet. [img]https://cdn.artofproblemsolving.com/attachments/d/f/8e363b40654ad0d8e100eac38319ee3784a7a7.png[/img]