Given is a triangle $ABC$ with median $BM$. The point $D$ lies on the line $AC$ after $C$, such that $BD=2CD$. The circle $(BMC)$ meets the segment $BD$ at $N$. Show that $AC+BM>2MN$.

Let $n\geq 3$ be a positive integer. Inside a $n\times n$ array there are placed $n^2$ positive numbers with sum $n^3$. Prove that we can find a square $2\times 2$ of 4 elements of the array, having the sides parallel with the sides of the array, and for which the sum of the elements in the square is greater than $3n$. [i]Radu Gologan[/i]

Determine all positive integers $d,$ such that there exists an integer $k\geq 3,$ such that One can arrange the numbers $d,2d,\ldots,kd$ in a row, such that the sum of every two consecutive of them is a perfect square.

Let $\alpha$ be the positive root of the quadratic equation $x^2 = 1990x + 1$. For any $m, n \in \mathbb N$, define the operation $m*n = mn + [\alpha m][ \alpha n]$, where $[x]$ is the largest integer no larger than $x$. Prove that $(p*q)*r = p*(q*r)$ holds for all $p, q, r \in \mathbb N.$

Given real numbers $x_1,x_2,y_1,y_2,z_1,z_2$ satisfying $x_1>0,x_2>0,x_1y_1>z_1^2$, and $x_2y_2>z_2^2$, prove that: \[ {8\over(x_1+x_2)(y_1+y_2)-(z_1+z_2)^2}\le{1\over x_1y_1-z_1^2}+{1\over x_2y_2-z_2^2}. \] Give necessary and sufficient conditions for equality.

A rectangle with integer side lengths has the property that its area minus $5$ times its perimeter equals $2023$. Find the minimum possible perimeter of this rectangle.

Let $f$ be a bijection of the set of all natural numbers on to itself. Prove that there exists positive integers $a < a+d < a+ 2d$ such that $f(a) < f(a+d) <f(a+2d)$

Show that if $ f(x)$ is a real-valued, continuous function on the half-line $ 0\leq x < \infty$, and \[ \int_0^{\infty} f^2(x)dx <\infty\] then the function \[ g(x)\equal{}f(x)\minus{}2e^{\minus{}x}\int_0^x e^tf(t)dt\] satisfies \[ \int _0^{\infty}g^2(x)dx\equal{}\int_0^{\infty}f^2(x)dx.\] [B. Szokefalvi-Nagy]

Find all $x$ for which, for all $n,$ \[\sum^n_{k=1} \sin {k x} \leq \frac{\sqrt{3}}{2}.\]

Let $n$ be a natural number such that $n!$ is a multiple of $2023$ and is not divisible by $37$. Find the largest power of $11$ that divides $n!$.

Determine all pairs of integers $(x, y)$ satisfying the equation \[y(x + y) = x^3- 7x^2 + 11x - 3.\]

Suppose that $ a_1$, $ a_2$, $ \ldots$, $ a_n$ are integers such that $ n\mid a_1 \plus{} a_2 \plus{} \ldots \plus{} a_n$. Prove that there exist two permutations $ \left(b_1,b_2,\ldots,b_n\right)$ and $ \left(c_1,c_2,\ldots,c_n\right)$ of $ \left(1,2,\ldots,n\right)$ such that for each integer $ i$ with $ 1\leq i\leq n$, we have \[ n\mid a_i \minus{} b_i \minus{} c_i \] [i]Proposed by Ricky Liu & Zuming Feng, USA[/i]

Points $M$ and $N$ are chosen on the sides $AB$ and BC of a triangle $ABC$. The segments $AN$ and $CM$ meet at $O$ such that $AO =CO$. Is the triangle $ABC$ necessarily isosceles, if (a) $AM = CN$? (b) $BM = BN$?

Let there be a given parabola $y^2=4ax$ in the coordinate plane. Consider all chords of the parabola that are visible at a right angle from the origin of the coordinate system. Prove that all these chords pass through a fixed point.

Given a number $b>0$, find all functions $f: \mathbb{R}\rightarrow\mathbb{R}$ such that: $f(x+y)=f(x).3^{b^{y}+f(y)-1}+b^{x}.\left(3^{b^{y}+f(y)-1}-b^{y}\right) \forall x,y\in\mathbb{R}$

Three identical spheres fit into a glass with rectangular sides and bottom and top in the form of regular hexagons such that every sphere touches every side of the glass. The glass has volume $108$ cm$^3$. What is the sidelength of the bottom? [img]https://1.bp.blogspot.com/-hBkYrORoBHk/XzcDt7B83AI/AAAAAAAAMXs/P5PGKTlNA7AvxkxMqG-qxqDVc9v9cU0VACLcBGAsYHQ/s0/2000%2BMohr%2Bp2.png[/img]

Two players take turns placing the numbers $1, 2, 3,. . . , 24$, in each of the $24$ squares on the surface of a $2 \times 2 \times 2$ cube (each number can be placed once). The second player wants the sum of the numbers in each cell the rings of $8$ cells encircling the cube were identical. Will he be able to the first player to stop him?

There are $n$ hoops on a circle. Rik numbers all hoops with a natural number so that all numbers from $1$ to $n$ occur exactly once. Then he makes one walk from hoop to hoop. He starts in hoop $1$ and then follows the following rule: if he gets to hoop $k$, then he walks to the hoop that places $k$ clockwise without getting into the intermediate hoops. The walk ends when Rik has to walk to a hoop he has already been to. The length of the walk is the number of hoops he passed on the way. For example, for $n = 6$ Rik can take a walk of length $5$ as the hoops are numbered as shown in the figure. [img]https://cdn.artofproblemsolving.com/attachments/2/a/3d4b7edbba4d145c7e00368f9b794f39572dc5.png[/img] (a) Determine for every even $n$ how Rik can number the hoops so that he has one walk of length $n$. (b) Determine for every odd $n$ how Rik can number the hoops so that he has one walk of length $n - 1$. (c) Show that for an odd $n$ there is no such numbering of the hoops that Rik can make a walk of length $n$.

In a triangle $ABC$ denote by $D,E,F$ the points where the angle bisectors of $\angle CAB,\angle ABC,\angle BCA$ respectively meet it's circumcircle. a) Prove that the orthocenter of triangle $DEF$ coincides with the incentre of triangle $ABC$. b) Prove that if $\overrightarrow{AD}+\overrightarrow{BE}+\overrightarrow{CF}=0$, then the triangle $ABC$ is equilateral. [i]Marin Ionescu[/i]

What is the value of $9901\cdot101-99\cdot10101?$ $\textbf{(A) }2\qquad\textbf{(B) }20\qquad\textbf{(C) }21\qquad\textbf{(D) }200\qquad\textbf{(E) }2020$

Let $B$ and $D$ be points on segments $[AE]$ and $[AF]$ respectively. Excircles of triangles $ABF$ and $ADE$ touching sides $BF$ and $DE$ is the same, and its center is $I$. $BF$ and $DE$ intersects at $C$. Let $P_1, P_2, P_3, P_4, Q_1, Q_2, Q_3, Q_4$ be the circumcenters of triangles $IAB, IBC, ICD, IDA, IAE, IEC, ICF, IFA$ respectively. [b]a) [/b] Show that points $P_1, P_2, P_3, P_4$ concylic and points $Q_1, Q_2, Q_3, Q_4$ concylic. [b]b) [/b] Denote centers of theese circles as $O_1$ and $O_2$. Prove that $O_1, O_2$ and $I$ are collinear.

The sweeties shop called "Olympiad" sells boxes of $6,9$ or $20$ chocolates. Groups of students from a school that is near the shop collect money to buy a chocolate for each student; to make this they buy a box and than give to everybody a chocolate. Like this students can create groups of $15=6+9$ students, $38=2*9+20$ students, etc. The seller has promised to the students that he can satisfy any group of students, and if he will need to open a new box of chocolate for any group (like groups of $4,7$ or $10$ students) than he will give all the chocolates for free to this group. Can there be constructed the biggest group that profits free chocolates, and if so, how many students are there in this group?

Let $a,b,c$ be non-negative real numbers such that \[b+c\leqslant a+1,\quad c+a\leqslant b+1,\quad a+b\leqslant c+1.\]Prove that $a^2+b^2+c^2\leqslant 2abc+1.$

$ABCDE$ is a convex pentagon such that the triangles $ABC, BCD, CDE, DEA$ and $EAB$ have equal areas. Show that $(1/4)$ area $(ABCDE) <$ area $(ABC) < (1/3)$ area $(ABCDE)$.

Let $ a$, $ b$, $ c$ be positive real numbers such that $ abc \equal{} 1$. Prove that \[ \frac {1}{a^{3}\left(b \plus{} c\right)} \plus{} \frac {1}{b^{3}\left(c \plus{} a\right)} \plus{} \frac {1}{c^{3}\left(a \plus{} b\right)}\geq \frac {3}{2}. \]