Given $10$ points in the space such that each $4$ points are not lie on a plane. Connect some points with some segments such that there are no triangles or quadrangles. Find the maximum number of the segments.

Let F be the midpoint of side BC or triangle ABC. Construct isosceles right triangles ABD and ACE externally on sides AB and AC with the right angles at D and E respectively. Show that DEF is an isosceles right triangle.

Triangle $ABC$ is given. Prove that $\frac{R}{r} > \frac{a}{h}$, where $R$ is the radius of the circumscribed circle, $r$ is the radius of the inscribed circle, $a$ is the length of the longest side, $h$ is the length of the shortest altitude.

Let $ACDB$ be a cyclic quadrilateral with circumcenter $\omega$. Let $AC=5$, $CD=6$, and $DB=7$. Suppose that there exists a unique point $P$ on $\omega$ such that $\overline{PC}$ intersects $\overline{AB}$ at a point $P_1$ and $\overline{PD}$ intersects $\overline{AB}$ at a point $P_2$, such that $AP_1=3$ and $P_2B=4$. Let $Q$ be the unique point on $\omega$ such that $\overline{QC}$ intersects $\overline{AB}$ at a point $Q_1$, $\overline{QD}$ intersects $\overline{AB}$ at a point $Q_2$, $Q_1$ is closer to $B$ than $P_1$ is to $B$, and $P_2Q_2=2$. The length of $P_1Q_1$ can be written as $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.

The diagram shows some squares whose sides intersect other squares at the midpoints of their sides. The shaded region has total area $7$. Find the area of the largest square. [img]https://cdn.artofproblemsolving.com/attachments/3/a/c3317eefe9b0193ca15f36599be3f6c22bb099.png[/img]

In right triangle $\triangle{ABC}$, a square $WXYZ$ is inscribed such that vertices $W$ and $X$ lie on hypotenuse $\overline{AB}$, vertex $Y$ lies on leg $\overline{BC}$, and vertex $Z$ lies on leg $\overline{CA}$. Let $\overline{AY}$ and $\overline{BZ}$ intersect at some point $P$. If the length of each side of square $WXYZ$ is $4$, the length of the hypotenuse $\overline{AB}$ is $60$, and the distance between point $P$ and point $G$, where $G$ denotes the centroid of $\triangle{ABC}$, is $\tfrac{a}{b}$, compute the value of $a+b$.

Prove that the simultaneous equations $$x^4 -x^2 =y^4 -y^2 =z^4 -z^2$$ are satisfied by the points of $4$ straight lines and $6$ ellipses, and by no other points.

An equilateral triangle $ABC$ is divided into $100$ congruent equilateral triangles. What is the greatest number of vertices of small triangles that can be chosen so that no two of them lie on a line that is parallel to any of the sides of the triangle $ABC$?

Circles $P$, $Q$, and $R$ are externally tangent to one another. The external tangent of $P$ and $Q$ that does not intersect $R$ intersects $P$ and $Q$ at $P_Q$ and $Q_P$ , respectively. $Q_R$,$R_Q$,$R_P$ , and $P_R$ are defined similarly. If the radius of $Q$ is $4$ and $\overline{Q_PP_Q} \parallel \overline{R_QQ_R}$, compute $R_PP_R$.

For $2k$ real numbers $a_1, a_2, ..., a_k$, $b_1, b_2, ..., b_k$ define a sequence of numbers $X_n$ by \[ X_n = \sum_{i=1}^k [a_in + b_i] \quad (n=1,2,...). \] If the sequence $X_N$ forms an arithmetic progression, show that $\textstyle\sum_{i=1}^k a_i$ must be an integer. Here $[r]$ denotes the greatest integer less than or equal to $r$.

Students in a school are arranged in an order that when you count from left to right, there will be $n$ students in the first row, $n-1$ students in the second row, $n - 2$ students in the third row,... until there is one student in the $n$th row. All the students face to the first row. For example, here is an arrangement for $n = 5$, where each $*$ represents one student: $*$ $* *$ $* * *$ $* * * *$ $* * * * *$ (first row) Each student will pick one of two following statement (except the student standing at the beginning of the row): i) The guy before me is telling the truth, while the guy standing next to him on the left is lying. ii) The guy before me is lying, while the guy standing next to him on the left is telling the truth. For $n = 2015$, find the maximum number of students telling the truth. (A student is lying if what he said is not true. Otherwise, he is telling the truth.)

In a regular $ 6n\plus{}1$-gon, $ k$ vertices are painted in red and the others in blue. Prove that the number of isosceles triangles whose vertices are of the same color does not depend on the arrangement of the red vertices.

We are given a cyclic quadrilateral $ABCD$ with a point $E$ on the diagonal $AC$ such that $AD=AE$ and $CB=CE$. Let $M$ be the center of the circumcircle $k$ of the triangle $BDE$. The circle $k$ intersects the line $AC$ in the points $E$ and $F$. Prove that the lines $FM$, $AD$ and $BC$ meet at one point. [i](4th Middle European Mathematical Olympiad, Individual Competition, Problem 3)[/i]

Lazim rolls two $24$-sided dice. From the two rolls, Lazim selects the die with the highest number. $N$ is an integer not greater than $24$. What is the largest possible value for $N$ such that there is a more than $50$% chance that the die Lazim selects is larger than or equal to $N$?

A hundred tourists arrive to a hotel at night. They know that in the hotel there are single rooms numbered as $1, 2, \ldots , n$, and among them $k{}$ (the tourists do not know which) are under repair, the other rooms are free. The tourists, one after another, check the rooms in any order (maybe different for different tourists), and the first room not under repair is taken by the tourist. The tourists don’t know whether a room is occupied until they check it. However it is forbidden to check an occupied room, and the tourists may coordinate their strategy beforehand to avoid this situation. For each $k{}$ find the smallest $n{}$ for which the tourists may select their rooms for sure. [i]Fyodor Ivlev[/i]

Assume that $P(x)$ is a polynomial with integer non negative coefficients, different from constant. Baron Munchausen claims that he can restore $P(x)$ provided he knows the values of $P(2)$ and $P(P(2))$ only. Is the baron's claim valid?