Give a geometric proof of the statement that the fold line on a sheet of paper is straight.

The points $E$ and $D$ lie in the interior of sides $AC$ and $BC$, respectively, of a triangle $ABC$. Let $F$ be the intersection of the lines $AD$ and $BE$.Show that the area of the traingles $ABC$ and $ABF$ satisfies: $ \frac{S_{ABC}}{S_{ABF}} = \frac{\mid{AC}\mid}{\mid{AE} \mid} + \frac{\mid{BC}\mid}{\mid{BD}\mid} - 1$.

Show that if the sides $a, b, c$ of a triangle satisfy the equation \[2(ab^2 + bc^2 + ca^2) = a^2b + b^2c + c^2a + 3abc,\] then the triangle is equilateral. Show also that the equation can be satisfied by positive real numbers that are not the sides of a triangle.

Let $A_1A_2A_3A_4$ be a tetrahedron. Consider the sphere centered at $A_1$ which is tangent to the face $A_2A_3A_4$ of the tetrahedron. Show that the surface area of the part of the sphere which is inside the tetrahedron is less than the area of the triangle $A_2A_3A_4$. [i]Sorin Rădulescu[/i]

Suppose $ H$ and $ O$ are the orthocenter and the circumcenter of acute triangle $ ABC$; $ AA_1$, $ BB_1$ and $ CC_1$ are the altitudes of the triangle. Point $ C_2$ is the reflection of $ C$ in $ A_1B_1$. Prove that $ H$, $ O$, $ C_1$ and $ C_2$ are concyclic.

In the interior of a cube we consider $\displaystyle 2003$ points. Prove that one can divide the cube in more than $\displaystyle 2003^3$ cubes such that any point lies in the interior of one of the small cubes and not on the faces.

On the sides $ BC $, $ CA $, $ AB $ of the triangle $ ABC $, the points $ D $, $ E $, $ F $ are chosen respectively, such that $$ \frac{|DB|}{|DC|} = \frac{|EC|}{|EA|} = \frac{|FA|}{|FB|}$$ Prove that if the triangle $ DEF $ is equilateral, then the triangle $ ABC $ is also equilateral.

Let $ABCD$ be a convex quadrilateral with $AB\le CD$. Points $E ,F$ are chosen on segment $AB$ and points $G ,H$ are chosen on the segment $CD$, are chosen such that $AE = BF = CG = DH <\frac{AB}{2}$. Let $P, Q$, and $R$ be the midpoints of $EG$, $FH$, and $CD$, respectively. It is known that $PR$ is parallel to $AD$ and $QR$ is parallel to $BC$. a) Show that $ABCD$ is a trapezoid. b) Let $d$ be the difference of the lengths of the parallel sides. Show that $2PQ\le d$.

Find all values of $ \alpha$ for which the curves $ y\equal{}\alpha x^2\plus{}\alpha x\plus{}\frac1{24}$ and $ x\equal{}\alpha y^2\plus{}\alpha y\plus{}\frac1{24}$ are tangent to each other.

We are given a convex polygon. Show that one can find a point $Q$ inside the polygon and three vertices $A_1,A_2,A_3$ (not necessarily consecutive) such that each ray $A_iQ$ ($i=1,2,3$) makes acute angles with the two sides emanating from $A_i$.

[b]p1.[/b] The entrance fee the county fair is $64$ cents. Unfortunately, you only have nickels and quarters so you cannot give them exact change. Furthermore, the attendent insists that he is only allowed to change in increments of six cents. What is the least number of coins you will have to pay? [b]p2.[/b] At the county fair, there is a carnival game set up with a mouse and six cups layed out in a circle. The mouse starts at position $A$ and every ten seconds the mouse has equal probability of jumping one cup clockwise or counter-clockwise. After a minute if the mouse has returned to position $A$, you win a giant chunk of cheese. What is the probability of winning the cheese? [b]p3.[/b] A clown stops you and poses a riddle. How many ways can you distribute $21$ identical balls into $3$ different boxes, with at least $4$ balls in the first box and at least $1$ ball in the second box? [b]p4.[/b] Watch out for the pig. How many sets $S$ of positive integers are there such that the product of all the elements of the set is $125970$? [b]p5.[/b] A good word is a word consisting of two letters $A$, $B$ such that there is never a letter $B$ between any two $A$'s. Find the number of good words with length $8$. [b]p6.[/b] Evaluate $\sqrt{2 -\sqrt{2 +\sqrt{2-...}}}$ without looking. [b]p7.[/b] There is nothing wrong with being odd. Of the first $2006$ Fibonacci numbers ($F_1 = 1$, $F_2 = 1$), how many of them are even? [b]p8.[/b] Let $f$ be a function satisfying $f (x) + 2f (27- x) = x$. Find $f (11)$. [b]p9.[/b] Let $A$, $B$, $C$ denote digits in decimal representation. Given that $A$ is prime and $A -B = 4$, nd $(A,B,C)$ such that $AAABBBC$ is a prime. [b]p10.[/b] Given $\frac{x^2+y^2}{x^2-y^2} + \frac{x^2-y^2}{x^2+y^2} = k$ , find $\frac{x^8+y^8}{x^8-y^8}$ in term of $k$. [b]p11.[/b] Let $a_i \in \{-1, 0, 1\}$ for each $i = 1, 2, 3, ..., 2007$. Find the least possible value for $\sum^{2006}_{i=1}\sum^{2007}_{j=i+1} a_ia_j$. [b]p12.[/b] Find all integer solutions $x$ to $x^2 + 615 = 2^n$ for any integer $n \ge 1$. [b]p13.[/b] Suppose a parabola $y = x^2 - ax - 1$ intersects the coordinate axes at three points $A$, $B$, and $C$. The circumcircle of the triangle $ABC$ intersects the $y$ - axis again at point $D = (0, t)$. Find the value of $t$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

In a triangle $ABC$ with $AB < BC < AC$, points $A' ,B' ,C'$ are such that $AA' \perp BC$ and $AA' = BC, BB' \perp CA$ and $BB'=CA$, and $CC' \perp AB$ and $CC'= AB$, as shown on the picture. Suppose that $\angle AC'B$ is a right angle. Prove that the points $A',B' ,C' $ are collinear.

[b]6.1 [/b] Two people went from point A to point B. The first one walked along highway at a speed of 5 km/h, and the second along a path at a speed of 4 km/h. The first of them arrived at point B an hour later and traveled 6 kilometers more. Find the distance from A to B along the highway. [b]6.2.[/b] A pedestrian walks along the highway at a speed of 5 km/hour. Along this highway in both directions at the same speed Buses run, meeting every 5 minutes. At 12 o'clock the pedestrian noticed that the buses met near him and, Continuing to walk, he began to count those oncoming and overtaking buses. At 2 p.m., buses met near him again. It turned out that during this time the pedestrian encountered 4 buses more than overtook him. Find the speed of the bus [b]6.3. [/b] Prove that the difference $43^{43} - 17^{17}$ is divisible by $10$. [b]6.4. [/b] Two squares are cut out of the chessboard on the border of the board. When is it possible and when is it not possible to cover with the remaining squares of the board? shapes of the view without overlay? [b]6.5.[/b] The distance from city A to city B (by air) is 30 kilometers, from B to C - 80 kilometers, from C to D - 236 kilometers, from D to E - 86 kilometers, from E to A- 40 kilometers. Find the distance from E to C. [b]6.6.[/b] Is it possible to write down the numbers from $ 1$ to $1963$ in a series so that any two adjacent numbers and any two numbers located one after the other were mutually prime? PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3983460_1963_leningrad_math_olympiad]here[/url].

Point $P$ lies on the side $CD$ of the cyclic quadrilateral $ABCD$ such that $\angle CBP = 90^{\circ}$. Let $K$ be the intersection of $AC,BP$ such that $AK = AP = AD$. $H$ is the projection of $B$ onto the line $AC$. Prove that $\angle APH = 90^{\circ}$. [i]Proposed by Iman Maghsoudi - Iran[/i]

Let ${\cal C}_1$ and ${\cal C}_2$ be concentric circles, with ${\cal C}_2$ in the interior of ${\cal C}_1$. From a point $A$ on ${\cal C}_1$ one draws the tangent $AB$ to ${\cal C}_2$ ($B\in {\cal C}_2$). Let $C$ be the second point of intersection of $AB$ and ${\cal C}_1$, and let $D$ be the midpoint of $AB$. A line passing through $A$ intersects ${\cal C}_2$ at $E$ and $F$ in such a way that the perpendicular bisectors of $DE$ and $CF$ intersect at a point $M$ on $AB$. Find, with proof, the ratio $AM/MC$.

An ant walks on the interior surface of a cube, he moves on a straight line. If ant reaches to an edge the he moves on a straight line on cube's net. Also if he reaches to a vertex he will return his path. a) Prove that for each beginning point ant can has infinitely many choices for his direction that its path becomes periodic. b) Prove that if if the ant starts from point $A$ and its path is periodic, then for each point $B$ if ant starts with this direction, then his path becomes periodic.

It is given parallelogram $ABCD$. On it's sides $AB, BC, CD, DA$ are chosen points $E, F, G, H$ such that area of $EFGH$ is half of the area of $ABCD$. Show that at least one of the quadrilaterals $ABFH$ and $AEGD$ is parallelogram.

Let $ABCD$ be a convex quadrilateral with $\angle ABC>90$, $CDA>90$ and $\angle DAB=\angle BCD$. Denote by $E$ and $F$ the reflections of $A$ in lines $BC$ and $CD$, respectively. Suppose that the segments $AE$ and $AF$ meet the line $BD$ at $K$ and $L$, respectively. Prove that the circumcircles of triangles $BEK$ and $DFL$ are tangent to each other. $\emph{Slovakia}$

Let $ABCD$ be a square and $O$ is your center. Let $E,F,G,H$ points in the segments $AB,BC,CD,AD$ respectively, such that $AE = BF = CG = DH$. The line $OA$ intersects the segment $EH$ in the point $X$, $OB$ intersects $EF$ in the point $Y$, $OC$ intersects $FG$ in the point $Z$ and $OD$ intersects $HG$ in the point $W$. If the $(EFGH) = 1$. Find: $(ABCD) \times (XYZW)$ Note $(P)$ denote the area of the polygon $P$.

A triangle $ABC$ with $\beta= 45^o$ is given. There is a point $P$ on side $BC$, where $BP : PC =1 : 2$ (inner division) and $\angle APC = 60^o$. Someone claims that you can do it with elementary geometric theorems alone without using the plane trigonometry, the size of the angle $\gamma$ determine.

Let $ABC$ be an acute triangle where $AC > BC$. Let $T$ denote the foot of the altitude from vertex $C$, denote the circumcentre of the triangle by $O$. Show that quadrilaterals $ATOC$ and $BTOC$ have equal area.

$ABCD$ is a rectangle. Points $K, L, M, N$ are chosen on $AB, BC, CD, DA$ respectively so that $KL$ is parallel to $MN$, and $KM$ is perpendicular to $LN$. Show that the intersection of $KM$ and $LN$ lies on $BD$.

Given is an isosceles triangle $ABC$ with $CA=CB$ and angle bisector $BD$, $D \in AC$. The line through the center $O$ of $(ABC)$, perpendicular to $BD$, meets $BC$ at $E$. The line through $E$, parallel to $BD$, meets $AC$ at $F$. Prove that $CE=DF$.

$n$ vectors are on the plane. We can move each vector forward and backeard on the line that the vector is on it. If there are 2 vectors that their endpoints concide we can omit them and replace them with their sum (If their sum is nonzero). Suppose with these operations with 2 different method we reach to a vector. Prove that these vectors are on a common line

3. In acute-angled triangle ABC point D is the perpendicular projection of C on the side AB. Point E is the perpendicular projection of D on the side BC. Point F lies on the side DE and: $\frac{EF}{FD}=\frac{AD}{DB}$ Prove that $CF \bot AE$