Let $ABC$ be an acute triange with $B,C$ fixed. Let $D$ be the midpoint of $BC$ and $E,F$ be the foot of the perpendiculars from $D$ to $AB,AC$, respectively. a) Let $O$ be the circumcenter of triangle $ABC$ and $M=EF\cap AO, N=EF\cap BC$. Prove that the circumcircle of triangle $AMN$ passes through a fixed point; b) Assume that tangents of the circumcircle of triangle $AEF$ at $E,F$ are intersecting at $T$. Prove that $T$ is on a fixed line.

Let $O$ be the circumcenter of triangle $\triangle ABC$, where $\angle BAC=120^{\circ}$. The tangent at $A$ to $(ABC)$ meets the tangents at $B,C$ at $(ABC)$ at points $P,Q$ respectively. Let $H,I$ be the orthocenter and incenter of $\triangle OPQ$ respectively. Define $M,N$ as the midpoints of arc $\overarc{BAC}$ and $OI$ respectively, and let $MN$ meet $(ABC)$ again at $D$. Prove that $AD$ is perpendicular to $HI$.

Let $x=\frac{\displaystyle\sum_{n=1}^{44} \cos n^\circ}{\displaystyle \sum_{n=1}^{44} \sin n^\circ}.$ What is the greatest integer that does not exceed $100x$?

Consider a $100 \times 100$ table, and identify the cell in row $a$ and column $b$, $1 \leq a, b \leq 100$, with the ordered pair $(a, b)$. Let $k$ be an integer such that $51 \leq k \leq 99$. A $k$-knight is a piece that moves one cell vertically or horizontally and $k$ cells to the other direction; that is, it moves from $(a, b)$ to $(c, d)$ such that $(|a-c|, |b - d|)$ is either $(1, k)$ or $(k, 1)$. The $k$-knight starts at cell $(1, 1)$, and performs several moves. A sequence of moves is a sequence of cells $(x_0, y_0)= (1, 1)$, $(x_1, y_1), (x_2, y_2)$, $\ldots, (x_n, y_n)$ such that, for all $i = 1, 2, \ldots, n$, $1 \leq x_i , y_i \leq 100$ and the $k$-knight can move from $(x_{i-1}, y_{i-1})$ to $(x_i, y_i)$. In this case, each cell $(x_i, y_i)$ is said to be reachable. For each $k$, find $L(k)$, the number of reachable cells.

What is the value of the improper integral $ \int_0 ^ {\pi} \log (\sin (x)) dx$?

Find all 4-tuples $(a,b,c,n)$ of naturals such that $n^a + n^b = n^c$

Do there exist a rectangle that can be partitioned into a regular hexagon with side length $1$, and several right triangles with side lengths $1, \sqrt3 , 2$?

Anna and Lisa go for a bike ride. Anna's bike breaks down $30$ kilometers before their final destination. The two decide to complete the ride with Lisa's bike as follows: At the beginning, Anna is riding a bike and Lisa leaves. At some point, Anna gets off the bike, parks it on the side of the road and continues by foot. When Lisa gets to the bike, she takes it and rides until she catches up with Anna. After that, they repeat the same procedure. We don't know how many times the procedure is repeated, but they arrive at the final goal at the same time. Anna walks at a speed of $4$ km/h and cycles at a speed of $15$ km/h. Lisa walks at $5$ km/h and cycles with $20$ km/h. How long does it take them to cover the last $30$ km of the road? (Neglect the time it takes to park, lock, unlock the bike, etc.)

Find the number of positive integers less than 101 that [i]can not [/i] be written as the difference of two squares of integers.

Let $\{\omega_1,\omega_2,\cdots,\omega_{100}\}$ be the roots of $\frac{x^{101}-1}{x-1}$ (in some order). Consider the set $$S=\{\omega_1^1,\omega_2^2,\omega_3^3,\cdots,\omega_{100}^{100}\}.$$ Let $M$ be the maximum possible number of unique values in $S,$ and let $N$ be the minimum possible number of unique values in $S.$ Find $M-N.$

Find the sum of $1\cdot 1!+2\cdot 2!+3\cdot 3!+\cdots+(n-1)(n-1)!+n\cdot n!$, where $n!=n(n-1)(n-2)\cdots2\cdot1$.

A subset $W$ of the set of real numbers is called a ring if it contains $1$ and if for all $a,b\in W$, the numbers $a-b$ and $ab$ are also in $W$. Let $S=\left\{\frac m{2^n}|m,n\in\mathbb Z\right\}$ and $T=\left\{\frac pq|p,q\in\mathbb Z,q\text{ odd}\right\}$. Then $\textbf{(A)}~\text{neither }S\text{ nor }T\text{ is a ring}$ $\textbf{(B)}~S\text{ is a ring, }T\text{ is not a ring}$ $\textbf{(C)}~T\text{ is a ring, }S\text{ is not a ring}$ $\textbf{(D)}~\text{both }S\text{ and }T\text{ are rings}$

Solve the system of equations: $ \begin{matrix} |x+y| + |1-x| = 6 \\ |x+y+1| + |1-y| = 4. \end{matrix} $

We say that a finite set $\mathcal{S}$ of points in the plane is [i]balanced[/i] if, for any two different points $A$ and $B$ in $\mathcal{S}$, there is a point $C$ in $\mathcal{S}$ such that $AC=BC$. We say that $\mathcal{S}$ is [i]centre-free[/i] if for any three different points $A$, $B$ and $C$ in $\mathcal{S}$, there is no points $P$ in $\mathcal{S}$ such that $PA=PB=PC$. (a) Show that for all integers $n\ge 3$, there exists a balanced set consisting of $n$ points. (b) Determine all integers $n\ge 3$ for which there exists a balanced centre-free set consisting of $n$ points. Proposed by Netherlands

Let $ a_1,a_2,...,a_6$ be real numbers such that: $ a_1 \not \equal{} 0, a_1a_6 \plus{} a_3 \plus{} a_4 \equal{} 2a_2a_5 \ \mathrm{and}\ a_1a_3 \ge a_2^2$ Prove that $ a_4a_6\le a_5^2$. When does equality holds?

Let $ABCD$ be a tetrahedron with $BA \perp AC,DB \perp (BAC)$. Denote by $O$ the midpoint of $AB$, and $K$ the foot of the perpendicular from $O$ to $DC$. Suppose that $AC = BD$. Prove that $\frac{V_{KOAC}}{V_{KOBD}}=\frac{AC}{BD}$ if and only if $2AC \cdot BD = AB^2$.

[b]p1.[/b] Four positive numbers are placed at the vertices of a rectangle. Each number is at least as large as the average of the two numbers at the adjacent vertices. Prove that all four numbers are equal. [b]p2.[/b] The sum $498+499+500+501=1998$ is one way of expressing $1998$ as a sum of consecutive positive integers. Find all ways of expressing $1998$ as a sum of two or more consecutive positive integers. Prove your list is complete. [b]p3.[/b] An infinite strip (two parallel lines and the region between them) has a width of $1$ inch. What is the largest value of $A$ such that every triangle with area $A$ square inches can be placed on this strip? Justify your answer. [b]p4.[/b] A plane divides space into two regions. Two planes that intersect in a line divide space into four regions. Now suppose that twelve planes are given in space so that a) every two of them intersect in a line, b) every three of them intersect in a point, and c) no four of them have a common point. Into how many regions is space divided? Justify your answer. [b]p5.[/b] Five robbers have stolen $1998$ identical gold coins. They agree to the following: The youngest robber proposes a division of the loot. All robbers, including the proposer, vote on the proposal. If at least half the robbers vote yes, then that proposal is accepted. If not, the proposer is sent away with no loot and the next youngest robber makes a new proposal to be voted on by the four remaining robbers, with the same rules as above. This continues until a proposed division is accepted by at least half the remaining robbers. Each robber guards his best interests: He will vote for a proposal if and only if it will give him more coins than he will acquire by rejecting it, and the proposer will keep as many coins for himself as he can. How will the coins be distributed? Explain your reasoning. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

An integer is written in each cell of a board of$ N$ rows and $N + 1$ columns. Prove that some columns (possibly none) can be deleted so that in each row the sum of the numbers left uncrossed out is even.

Show that there exists no function $f:\mathbb {R}\rightarrow \mathbb {R}$ that satisfies $f(f(x))-x^2+x+3=0$ for arbitrary real variable $x$. (Same as KMO 2004 P1)

The Wonder Island Intelligence Service has $16$ spies in Tartu. Each of them watches on some of his colleagues. It is known that if spy $A$ watches on spy $B$, then $B$ does not watch on $A$. Moreover, any $10$ spies can numbered in such a way that the first spy watches on the second, the second watches on the third and so on until the tenth watches on the first. Prove that any $11$ spies can also be numbered is a similar manner.

Prove or disprove: there is at least one straight line normal to the graph of $y=\cosh x$ at a point $(a,\cosh a)$ and also normal to the graph of $y=$ $\sinh x$ at a point $(c,\sinh c).$

There are three different points on the line $ A $, $ B $, $ C $ and a point $ S $ outside this line; perpendicularly drawn at points $ A $, $ B $, $ C $ to the lines $ SA $, $ SB $, $ SC $ intersect at points $ M $, $ N $, $ P $. Prove that the points $ M $, $ N $, $ P $, $ S $ lie on the circle

Can you mark the cube's vertices with the three-digit binary numbers in such a way, that the numbers at all the possible couples of neighbouring vertices differ in at least two digits?

Let $n$ be a positive integer. Show that there are positive real numbers $a_0, a_1, \dots, a_n$ such that for each choice of signs the polynomial $$\pm a_nx^n\pm a_{n-1}x^{n-1} \pm \dots \pm a_1x \pm a_0$$ has $n$ distinct real roots. (Proposed by Stephan Neupert, TUM, München)

A circle of radius 1 is located in a right-angled trihedron and touches all its faces. Find the locus of centers of such circles.