We are given $3n$ points $A_1,A_2, \ldots , A_{3n}$ in the plane, no three of them collinear. Prove that one can construct $n$ disjoint triangles with vertices at the points $A_i.$

Bob and Alice each have a bag that contains one ball of each of the colors blue, green, orange, red, and violet. Alice randomly selects one ball from her bag and puts it into Bob's bag. Bob then randomly selects one ball from his bag and puts it into Alice's bag. What is the probability that after this process, the contents of the two bags are the same? $ \textbf{(A) } \frac 1{10} \qquad \textbf{(B) } \frac 16 \qquad \textbf{(C) } \frac 15 \qquad \textbf{(D) } \frac 13 \qquad \textbf{(E) } \frac 12$

Determine the set of all real numbers $r$ for which there exists an infinite sequence $a_1,a_2,\dots$ of positive integers satisfying the following three properties: (1) No number occurs more than once in the sequence. (2) The sum of two different elements of the sequence is never a power of two. (3) For all positive integers $n$, we have $a_n<r \cdot n$.

Farmer Pythagoras has a field in the shape of a right triangle. The right triangle's legs have lengths 3 and 4 units. In the corner where those sides meet at a right angle, he leaves a small unplanted square $S$ so that from the air it looks like the right angle symbol. The rest of the field is planted. The shortest distance from $S$ to the hypotenuse is 2 units. What fraction of the field is planted? [asy] draw((0,0)--(4,0)--(0,3)--(0,0)); draw((0,0)--(0.3,0)--(0.3,0.3)--(0,0.3)--(0,0)); fill(origin--(0.3,0)--(0.3,0.3)--(0,0.3)--cycle, gray); label("$4$", (2,0), N); label("$3$", (0,1.5), E); label("$2$", (.8,1), E); label("$S$", (0,0), NE); draw((0.3,0.3)--(1.4,1.9), dashed); [/asy] $\textbf{(A) } \frac{25}{27} \qquad \textbf{(B) } \frac{26}{27} \qquad \textbf{(C) } \frac{73}{75} \qquad \textbf{(D) } \frac{145}{147} \qquad \textbf{(E) } \frac{74}{75} $

$ABCD$ - convex quadrilateral. $\angle A+ \angle D=150, \angle B<150, \angle C<150$ Prove, that area $ABCD$ is greater than $\frac{1}{4}(AB*CD+AB*BC+BC*CD)$

Functions f and g are defined on the whole real line and are mutually inverse: g(f(x))=x, f(g(y))=y for all x, y. It is known that f can be written as a sum of periodic and linear functions: f(x)=kx+h(x) for some number k and a periodic function h(x). Show that g can also be written as a sum of periodic and linear functions. (A functions h(x) is called periodic if there exists a non-zero number d such that h(x+d)=h(x) for any x.)

Let $A,B$ be two points interior of circle $C(O,R)$ and $M$ a point on the circle. Let $A_1,B_1$ be the intersections of the circle with lines $MA$,$MB$ respectively. Let $G$ be the midpoint of $AB$and $G_1= C\cap MG$. Prove that$$\frac{MA}{AA_1}+ \frac{MB}{BB_1}> 2\frac{MG}{GG_1}$$

Find the minimum value of $m + n$, where $m$ and $n$ are positive integers satisfying: $2023 \vert m + 2025n$ $2025 \vert m + 2023n$ [i]Proposed by Prudencio Guerrero Fernández [/i]

A prime $p>3$ is given. Let $K$ be the number of such permutations $(a_1, a_2, \ldots, a_p)$ of $\{ 1, 2, \ldots, p\}$ such that $$a_1a_2+a_2a_3+\ldots + a_{p-1}a_p+a_pa_1$$ is divisible by $p$. Prove $K+p$ is divisible by $p^2$.

Let $\vartriangle ABC$ be a right triangle with $\angle B = 90^o$ and $|AB| > |BC|$, and let $\Gamma$ be the semicircle with diameter $AB$ that lies on the same side as $C$. Let $P$ be a point on $\Gamma$ such that $|BP| = |BC|$ and let $Q$ be on $AB$ such that $|AP| = |AQ|$. Prove that the midpoint of $CQ$ lies on $\Gamma$.

Solve the equation $\sqrt{1981-\sqrt{1996+x}}=x+15$

The work team was working at a rate fast enough to process $1250$ items in ten hours. But after working for six hours, the team was given an additional $150$ items to process. By what percent does the team need to increase its rate so that it can still complete its work within the ten hours?

Let $a$ be a positive integer. How many non-negative integer solutions x does the equation $\lfloor \frac{x}{a}\rfloor = \lfloor \frac{x}{a+1}\rfloor$ have? $\lfloor ~ \rfloor$ ---> [url=http://en.wikipedia.org/wiki/Floor_function]Floor Function[/url].

Let $$A=\left( \begin{array}{cc} 4 & -\sqrt{5} \\ 2\sqrt{5} & -3 \end{array} \right) $$ Find all pairs of integers \(m,n\) with \(n \geq 1\) and \(|m| \leq n\) such as all entries of \(A^n-(m+n^2)A\) are integer.

$ (a)$ A group of people attends a party. Each person has at most three acquaintances in the group, and if two people do not know each other, then they have a common acquaintance in the group. What is the maximum possible number of people present? $ (b)$ If, in addition, the group contains three mutual acquaintances, what is the maximum possible number of people?

Place all numbers from 1 to 10 to the boxes such that every number except the uppermost is equal to the difference between the two numbers on its top. [asy] unitsize(-4); draw((0,0)--(5,0)--(5,5)--(0,5)--cycle); draw((10,0)--(15,0)--(15,5)--(10,5)--cycle); draw((20,0)--(25,0)--(25,5)--(20,5)--cycle); draw((30,0)--(35,0)--(35,5)--(30,5)--cycle); draw((5,10)--(10,10)--(10,15)--(5,15)--cycle); draw((15,10)--(20,10)--(20,15)--(15,15)--cycle); draw((25,10)--(30,10)--(30,15)--(25,15)--cycle); draw((10,20)--(15,20)--(15,25)--(10,25)--cycle); draw((20,20)--(25,20)--(25,25)--(20,25)--cycle); draw((15,30)--(20,30)--(20,35)--(15,35)--cycle); draw((2.5,5)--(7.5, 10)); draw((12.5,5)--(17.5, 10)); draw((22.5,5)--(27.5, 10)); draw((32.5,5)--(27.5, 10)); draw((22.5,5)--(17.5, 10)); draw((12.5,5)--(7.5, 10)); draw((7.5,15)--(12.5, 20)); draw((17.5,15)--(22.5, 20)); draw((27.5,15)--(22.5, 20)); draw((17.5,15)--(12.5, 20)); draw((12.5,25)--(17.5, 30)); draw((22.5,25)--(17.5, 30)); [/asy]The number in the lower box is at most $\textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ 5$

What is $1-2+3-4$? $\textbf{(A) } {-}2\qquad\textbf{(B) } {-}1\qquad\textbf{(C) } 1\qquad\textbf{(D) } 4\qquad\textbf{(E) } 9$

Let $O$ be the circumcenter of triangle $ABC,$ and $D$ be an arbitrary point on the circumcircle of $ABC.$ Let points $X, Y$ and $Z$ be the orthogonal projections of point $D$ onto lines $OA, OB$ and $OC,$ respectively. Prove that the incenter of triangle $XYZ$ is on the Simson-Wallace line of triangle $ABC$ corresponding to point $D.$

Let $BB_1$ and $CC_1$ be the altitudes of an acute-angled triangle $ABC$, which intersect its angle bisector $AL$ at two different points $P$ and $Q$, respectively. Denote by $F$ such a point that $PF\parallel AB$ and $QF\parallel AC$, and by $T$ the intersection point of the tangents drawn at points $B$ and $C$ to the circumscribed circle of the triangle $ABC$. Prove that the points $A, F$ and $T$ lie on the same line.

We want to choose telephone numbers for a city. The numbers have $10$ digits and $0$ isn’t used in the numbers. Our aim is: We don’t choose some numbers such that every $2$ telephone numbers are different in more than one digit OR every $2$ telephone numbers are different in a digit which is more than $1$. What is the maximum number of telephone numbers which can be chosen? In how many ways, can we choose the numbers in this maximum situation?

All the two-digit numbers from $19$ to $80$ are written in a line without spaces. Is the obtained number $192021....7980$ divisible by $1980$?

Show that $\frac{x^2}{1 - x}+\frac{(1 - x)^2}{x} \ge 1$ for all real numbers $x$, where $0 < x < 1$

Suppose $a,b,c \in \mathbb R^+$and \[\frac1{a^2+1}+\frac1{b^2+1}+\frac1{c^2+1}=2\] Prove that $ab+ac+bc\leq \frac32$

In a triangle $ABC$, it is known that $\angle A=100^{\circ}$ and $AB=AC$. The internal angle bisector $BD$ has length $20$ units. Find the length of $BC$ to the nearest integer, given that $\sin 10^{\circ} \approx 0.174$

Find all triples $(p,q,r)$ of pairwise distinct primes such that \[p\mid q+r, q\mid r+2p, r\mid p+3q.\]