Let $\triangle ABC$ be a triangle. Let $M$ be the midpoint of $BC$ and let $D$ be a point on the interior of side $AB$. The intersection of $AM$ and $CD$ is called $E$. Suppose that $|AD|=|DE|$. Prove that $|AB|=|CE|$.

Find all functions $f : R \to R$ such for any $x, y \in R,$ $$(x - y)f(x + y) - (x + y)f(x - y) = 4xy(x^2 - y^2)$$

A rectangle of paper of $3$ cm by $9$ cm is folded along a straight line, making two opposite vertices coincide. In this way a pentagon is formed. Calculate it's area.

In how many ways can $345$ be written as the sum of an increasing sequence of two or more consecutive positive integers? $\textbf{(A)}\ 1\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 5\qquad\textbf{(D)}\ 6\qquad\textbf{(E)}\ 7$

Find all the positive integers $p$ with the property that the sum of the first $p$ positive integers is a four-digit positive integer whose decomposition into prime factors is of the form $2^m3^n(m + n)$, where $m, n \in N^*$.

In a regular $2n$-gonal prism, bases $A_1A_2\cdots A_{2n}$ and $B_1B_2\cdots B_{2n}$ have circumradii equal to $R$. If the length of the lateral edge $A_1B_1$ varies, the angle between the line $A_1B_{n+1}$ and the plane $A_1A_3B_{n+2}$ is maximal for $A_1B_1=2R\cos\frac\pi{2n}$.

Find all functions $f:\mathbb R\to\mathbb R$ such that $$(x-y)f(x+y) - (x+y)f(x-y) = 2y(f(x)-f(y) - 1)$$for all $x, y\in\mathbb R$.

Hull Street is nine miles long. The first two miles of Hull Street are in Richmond, and the last two miles of the street are in Midlothian. If Kathy starts driving along Hull Street from a random point in Richmond and stops on the street at a random point in Midlothian, what is the probability that Kathy drove less than six miles? $\text{(A) }\frac{1}{16}\qquad\text{(B) }\frac{1}{9}\qquad\text{(C) }\frac{1}{8}\qquad\text{(D) }\frac{1}{6}\qquad\text{(E) }\frac{1}{4}$

Let $X$ and $Y$ be independent random variables with "Saint-Petersburg" distribution, i.e. for any $k=1,2,\ldots$ their value is $2^k$ with probability $\frac{1}{2^k}$. Show that $X$ and $Y$ can be realized on a sufficiently big probability space such that there exists another pair of independent "Saint-Petersburg" random variables $(X', Y')$ on this space with the property that $X+Y=2X'+Y'I(Y'\le X')$ almost surely (here $I(A)$ denotes the indicator function of the event $A$). (translated by L. Erdős)

For positive integer $n\geq 3$ and real numbers $a_1,...,a_n,b_1,...,b_n$, prove the following. $$\sum_{i=1}^n a_i(b_i-b_{i+3})\leq\frac{3n}{8}\sum_{i=1}^n((a_i-a_{i+1})^2+(b_i-b_{i+1})^2)$$ ($a_{n+1}=a_1$, and for $i=1,2,3$ $b_{n+i}=b_i$.)

Consider $27$ unit-cubes assembled into one $3 \times 3 \times 3$ cube. Let $A$ and $B$ be two opposite corners of this large cube. Remove the one unit-cube not visible from the exterior, along with all six unit-cubes in the center of each face. Compute the minimum distance an ant has to walk along the surface of the modified cube to get from $A$ to $B$. [img]https://cdn.artofproblemsolving.com/attachments/0/5/d3aa802eae40cfe717088445daabd5e7194691.png[/img]

Find all natural numbers $x,y,z$ such that \[(2^x-1)(2^y-1)=2^{2^z}+1.\]

Two circles meet at points $A$ and $B$. A line through $B$ intersects the first circle again at $K$ and the second circle at $M$. A line parallel to $AM$ is tangent to the first circle at $Q$. The line $AQ$ intersects the second circle again at $R$. $(a)$ Prove that the tangent to the second circle at $R$ is parallel to $AK$. $(b)$ Prove that these two tangents meet on $KM$.

Determine the largest possible constant \( C \) such that for any positive real numbers \( x, y, z \), which are the sides of a triangle, the following inequality holds: \[ \frac{xy}{x^2 + y^2 + xz} + \frac{yz}{y^2 + z^2 + yx} + \frac{zx}{z^2 + x^2 + zy} \geq C. \] [i]Proposed by Vadym Solomka[/i]

A circle $\mathcal C$ and five distinct points $M_1,M_2,M_3,M_4$ and $M$ on $\mathcal C$ are given in the plane. Prove that the product of the distances from $M$ to lines $M_1M_2$ and $M_3M_4$ is equal to the product of the distances from $M$ to the lines $M_1M_3$ and $M_2M_4$. What can one deduce for $2n+1$ distinct points $M_1,\ldots,M_{2n},M$ on $\mathcal C$?

In the tetrahedron $DABC, AB=BC, \angle DBC =\angle DBA$. Prove that $AC \perp DB$.

Let $ABC$ be an acute triangle with $AX, BY$ and $CZ$ as its altitudes. $\bullet$ Line $\ell_A$, which is parallel to $YZ$, intersects $CA$ at $A_1$ between $C$ and $A$, and intersects $AB$ at $A_2$ between $A$ and $B$. $\bullet$ Line $\ell_B$, which is parallel to $ZX$, intersects $AB$ at $B_1$ between $A$ and $B$, and intersects $BC$ at $B_2$ between $B$ and $C$. $\bullet$ Line $\ell_C$, which is parallel to $XY$ , intersects $BC$ at $C_1$ between $B$ and $C$, and intersects $CA$ at $C_2$ between $C$ and $A$. Suppose that the perimeters of the triangles $\vartriangle AA_1A_2$, $\vartriangle BB_1B_2$ and $\vartriangle CC_1C_2$ are equal to $CA+AB,AB +BC$ and $BC +CA$, respectively. Prove that $\ell_A, \ell_B$ and $\ell_C$ are concurrent.

Show that the number of integral-sided right triangles whose ratio of area to semi-perimeter is $p^{m}$, where $p$ is a prime and $m$ is an integer, is $m+1$ if $p=2$ and $2m+1$ if $p \neq 2$.

In a $4 \times 6$ grid, all edges and diagonals are drawn (see attachment). Determine the number of parallelograms in the grid that uses only the line segments drawn and none of its four angles are right.

The increasing interval of $f(x)=\log_{\frac{1}{2}}(x^2-2x-3)$ is $\text{(A)}(-\infty,-1)\qquad\text{(B)}(-\infty,1)\qquad\text{(C)}(1,+\infty)\qquad\text{(D)}(3,+\infty)$

Let $A_n$ be the area outside a regular $n$-gon of side length $1$ but inside its circumscribed circle, let $B_n$ be the area inside the $n$-gon but outside its inscribed circle. Find the limit as $n$ tends to infinity of $\dfrac{A_n}{B_n}$.

Find all functions $ f: \mathbb R\longrightarrow \mathbb R$ such that for each $ x,y\in\mathbb R$: \[ f(xf(y)) \plus{} y \plus{} f(x) \equal{} f(x \plus{} f(y)) \plus{} yf(x)\]

Let $AXBY$ be a convex quadrilateral. The incircle of $\triangle AXY$ has center $I_A$ and touches $\overline{AX}$ and $\overline{AY}$ at $A_1$ and $A_2$ respectively. The incircle of $\triangle BXY$ has center $I_B$ and touches $\overline{BX}$ and $\overline{BY}$ at $B_1$ and $B_2$ respectively. Define $P = \overline{XI_A} \cap \overline{YI_B}$, $Q = \overline{XI_B} \cap \overline{YI_A}$, and $R = \overline{A_1B_1} \cap \overline{A_2B_2}$. [list=a] [*] Prove that if $\angle AXB = \angle AYB$, then $P$, $Q$, $R$ are collinear. [*] Prove that if there exists a circle tangent to all four sides of $AXBY$, then $P$, $Q$, $R$ are collinear. [/list]

Prove that there is no bijective function $f : \left\{1,2,3,\ldots \right\}\rightarrow \left\{0,1,2,3,\ldots \right\}$ such that $f(mn)=f(m)+f(n)+3f(m)f(n)$.

Let $n \geq 1$ be an integer. A [b]path[/b] from $(0,0)$ to $(n,n)$ in the $xy$ plane is a chain of consecutive unit moves either to the right (move denoted by $E$) or upwards (move denoted by $N$), all the moves being made inside the half-plane $x \geq y$. A [b]step[/b] in a path is the occurence of two consecutive moves of the form $EN$. Show that the number of paths from $(0,0)$ to $(n,n)$ that contain exactly $s$ steps $(n \geq s \geq 1)$ is \[\frac{1}{s} \binom{n-1}{s-1} \binom{n}{s-1}.\]