Let $ABC$ be a triangle and let $P$ be a point in the plane of $ABC$ that is inside the region of the angle $BAC$ but outside triangle $ABC$. [b](a)[/b] Prove that any two of the following statements imply the third. [list] [b](i)[/b] the circumcentre of triangle $PBC$ lies on the ray $\stackrel{\to}{PA}$. [b](ii)[/b] the circumcentre of triangle $CPA$ lies on the ray $\stackrel{\to}{PB}$. [b](iii)[/b] the circumcentre of triangle $APB$ lies on the ray $\stackrel{\to}{PC}$.[/list] [b](b)[/b] Prove that if the conditions in (a) hold, then the circumcentres of triangles $BPC,CPA$ and $APB$ lie on the circumcircle of triangle $ABC$.

Determine all real values of parameter $p$ such that the equation \[|x-2|+|y-3|+y=p\] is an equation of a ray in the plane $xy.$

A sequence $(x_n)_{n= 1}^{\infty}$ satisfies $x_1 = 1$ and for each $n > 1, x_n = \pm (n-1)x_{n-1} \pm (n-2)x_{n-2} \pm ... \pm 2x_2 \pm x_1$. Prove that the signs ” $\pm$” can be chosen so that $x_n \ne 12$ holds only for finitely many $n$.

Prove that the equation $3y^2 = x^4 + x$ has no positive integer solutions.

Prove that there doesn’t exist any prime $p$ such that every power of $p$ is a palindrome (a palindrome is a number that is read the same from the left as it is from the right; in particular, a number that ends in one or more zeros cannot be a palindrome).

The following method of approximate measurement is known for distances. Suppose, for example, that the observer is on the river bank at point $C$ in order to measure its width. To do this, he fixes point $A$ on the opposite bank so that the angle between the shoreline and the line $CA$ is close to the line. Then the observer pulls forward the right hand with the raised thumb, closes left eye and aligns the raised finger with point $A$. Next, opens the left eye, closes right and estimates the distance between the point on the opposite bank to which the finger points, and point $A$. Multiply this distance by $10$ and get the approximate value of the distance to point $A$, ie the width of the river. Justify this method of measuring distance. [hide=original wording]Відомий наступний спосіб наближеного вимірювання відстані. Нехай, наприклад, спостерігач знаходиться на березі річки у точці C і має на меті виміряти її ширину. Для цього він фіксує точку A на протилежному березі так, щоб кут між лінією берега і прямою CA був близьким до прямого. Потім спостерігач витягує вперед праву руку з піднятим вгору великим пальцем, заплющує ліве око і суміщає піднятий палець з точкою A. Далі, відкриває ліве око, заплющує праве і оцінює відстань між точкою на протилежному березі, на яку вказує палець, і точкою A. Цю відстань множить на 10 і отримує наближене значення відстані до точки A, тобто ширини річки. Обґрунтуйте цей спосіб вимірювання відстані.[/hide]

The quadratic equation $ x^2 \plus{} mx \plus{} n \equal{} 0$ has roots that are twice those of $ x^2 \plus{} px \plus{} m \equal{} 0$, and none of $ m,n,$ and $ p$ is zero. What is the value of $ n/p$? $ \textbf{(A)}\ 1\qquad \textbf{(B)}\ 2\qquad \textbf{(C)}\ 4\qquad \textbf{(D)}\ 8\qquad \textbf{(E)}\ 16$

Given a circle $S$ and its $121$ chords $P_i (i=1,2,\ldots,121)$, each with a point $A_i(i=1,2,\ldots,121)$ on it. Prove that there exists a point $X$ on the circumference of $S$ such that: there exist $29$ distinct indices $1\le k_1\le k_2\le\ldots\le k_{29}\le 121$, such that the angle formed by ${A_{k_j}}X$ and ${P_{k_j}}$ is smaller than $21$ degrees for every $j=1,2,\ldots,29$.

Julia and Hansol are having a math-off. Currently, Julia has one more than twice as many points as Hansol. If Hansol scores $6$ more points in a row, he will tie Julia’s score. How many points does Julia have?

If three of the roots of $x^4+ax^2+bx+c=0$ are $1$, $2$, and $3$, then the value of $a+c$ is: $\text{(A)}\ 35 \qquad \text{(B)}\ 24\qquad \text{(C)}\ -12\qquad \text{(D)}\ -61 \qquad \text{(E)}\ -63$

Let $m,n$ be positive integers. Find the minimum positive integer $N$ which satisfies the following condition. If there exists a set $S$ of integers that contains a complete residue system module $m$ such that $| S | = N$, then there exists a nonempty set $A \subseteq S$ so that $n\mid {\sum\limits_{x \in A} x }$.

Let be a sequence of real numbers $ \left( x_n \right)_{n\geqslant 0} $ with $ x_0\neq 0,1 $ and defined as $ x_{n+1}=x_n+x_n^{-1/x_0} . $ [b]a)[/b] Show that the sequence $ \left( x_n\cdot n^{-\frac{x_0}{1+x_0}} \right)_{n\geqslant 0} $ is convergent. [b]b)[/b] Prove that $ \inf_{x_0\neq 0,1} \lim_{n\to\infty } x_n\cdot n^{-\frac{x_0}{1+x_0}} =1. $

Solve the equation $$2(x-6)=\dfrac{x^2}{(1+\sqrt{x+1})^2}$$

Show that there exists a $10 \times 10$ table of distinct natural numbers such that if $R_i$ is equal to the multiplication of numbers of row $i$ and $S_i$ is equal to multiplication of numbers of column $i$, then numbers $R_1$, $R_2$, ... , $R_{10}$ make a nontrivial arithmetic sequence and numbers $S_1$, $S_2$, ... , $S_{10}$ also make a nontrivial arithmetic sequence. (A nontrivial arithmetic sequence is an arithmetic sequence with common difference between terms not equal to $0$).

Let $n\geq 5$ an integer and consider a regular $n$-gon. Initially, Nacho is situated in one of the vertices of the $n$-gon, in which he puts a flag. He will start moving clockwise. First, he moves one position and puts another flag, then, two positions and puts another flag, etcetera, until he finally moves $n-1$ positions and puts a flag, in such a way that he puts $n$ flags in total. ¿For which values of $n$, Nacho will have put a flag in each of the $n$ vertices?

Let $n$ be a fixed positive integer. How many ways are there to write $n$ as a sum of positive integers, \[n = a_1 + a_2 + \cdots a_k\] with $k$ an arbitrary positive integer and $a_1 \le a_2 \le \cdots \le a_k \le a_1 + 1$? For example, with $n = 4$, there are four ways: $4$, $2 + 2$, $1 + 1 + 2$, $1 + 1 + 1 + 1$.

Compute $$\int \limits_{-\frac{\pi}{4}}^{\frac{\pi}{4}}\frac{\cos x+1-x^2}{(1+x\sin x)\sqrt{1-x^2}}dx$$

Let $A$ be the sum of three consecutive integers and $B$ be the sum of the exact three consecutive integers. Is it possible to have $AB=33333$ ?

Evaluate the integral $\int_0^{\pi/2} \sqrt{\tan \theta} d\theta$.

Find the minimum value of $x^2+2x+ \frac{24}{x}$ for $x > 0$.

Two circles $\gamma_1, \gamma_2$ in a plane, with centers $A$ and $B$ respectively, intersect at $C$ and $D$. Suppose that the circumcircle of $ABC$ intersects $\gamma_1$ in $E$ and $\gamma_2$ in $F$, where the arc $EF$ not containing $C$ lies outside $\gamma_1$ and $\gamma_2$. Prove that this arc $EF$ is bisected by the line $CD$.

Let $d$ be an even positive integer. John writes the numbers $1^2 ,3^2 ,\ldots,(2n-1)^2 $ on the blackboard and then chooses three of them, let them be ${a_1}, {a_2}, {a_3}$, erases them and writes the number $1+ \displaystyle\sum_{1\le i<j\leq 3} |{a_i} -{a_j}|$ He continues until two numbers remain written on on the blackboard. Prove that the sum of squares of those two numbers is different than the numbers $1^2 ,3^2 ,\ldots,(2n-1)^2$. (Albania)

Find all functions $f: \mathbb N \to \mathbb N$ such that for all positive integers $m,n$, [b](i)[/b] $mf(f(m))=\left( f(m) \right)^2$, [b](ii)[/b] If $\gcd(m,n)=d$, then $f(mn) \cdot f(d)=d \cdot f(m) \cdot f(n)$, [b](iii)[/b] $f(m)=m$ if and only if $m=1$.

In isoceles $\triangle ABC$, $AB=AC$, $I$ is its incenter, $D$ is a point inside $\triangle ABC$ such that $I,B,C,D$ are concyclic. The line through $C$ parallel to $BD$ meets $AD$ at $E$. Prove that $CD^2=BD\cdot CE$.

Let the real numbers be $a, b, c, d$ with $a \ge b$ and $c \ge d$. Prove that the equation $$(x + a) (x + d) + (x + b) (x + c) = 0$$ has real roots.