Let $ABC$ be a triangle and $I$ be its incenter. Let $D$ be the intersection of the exterior bisectors of $\angle BAC$ and $\angle BIC$, $E$ be the intersection of the exterior bisectors of $\angle ABC$ and $\angle AIC$, and $F$ be the intersection of the exterior bisectors of $\angle ACB$ and $\angle AIB$. Prove that $D$, $E$, $F$ are collinear

Let $ABCD$ be a quadrilateral with an inscribed circle $\omega$. Let $I$ be the center of $\omega$, and let $IA=12,$ $IB=16,$ $IC=14,$ and $ID=11$. Let $M$ be the midpoint of segment $AC$. Compute the ratio $\frac{IM}{IN}$, where $N$ is the midpoint of segment $BD$.

Find the range of the function $f: \mathbb{R} \to \mathbb{R},$ $$f(x)=\frac{3+2\sin x}{\sqrt{1+\cos x}+\sqrt{1-\cos x}}.$$

Prove that all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ satisfying, for all real $x$, \[ f(x)=f(2x)=f(1-x)\] are periodic.

[b]a)[/b] Provide an example of a sequence $ \left( a_n \right)_{n\ge 1} $ of positive real numbers whose series converges, and has the property that each member (sequence) of the family of sequences $ \left(\left( n^{\alpha } a_n \right)_{n\ge 1}\right)_{\alpha >0} $ is unbounded. [b]b)[/b] Let $ \left( b_n \right)_{n\ge 1} $ be a sequence of positive real numbers, having the property that $$ nb_{n+1}\leqslant b_1+b_2+\cdots +b_n, $$ for any natural number $ n. $ Prove that the following relations are equivalent: $\text{(i)} $ there exists a convergent member (series) of the family of series $ \left( \sum_{i=1}^{\infty } b_i^{\beta } \right)_{\beta >0} $ $ \text{(ii)} $ there exists a member (sequence) of the family of sequences $ \left(\left( n^{\beta } b_n \right)_{n\ge 1}\right)_{\beta >0} $ that is convergent to $ 0. $ [i]Eugen Păltănea[/i]

Given an equilateral triangle $ABC$ of side $a$ in a plane, let $M$ be a point on the circumcircle of the triangle. Prove that the sum $s = MA^4 +MB^4 +MC^4$ is independent of the position of the point $M$ on the circle, and determine that constant value as a function of $a$.

Given that $3 \sin x + 4 \cos x = 5$, where $x$ is in $(0, \frac{\pi}{2})$ , find $2 \sin x + \cos x + 4 \tan x$.

Suppose $50x$ is divisible by 100 and $kx$ is not divisible by 100 for all $k=1,2,\cdots, 49$ Find number of solutions for $x$ when $x$ takes values $1,2,\cdots 100$. [list=1] [*] 20 [*] 25 [*] 15 [*] 50 [/list]

Let p be a prime and $f: Z_p \to C$ a complex valued function defined on a cyclic group of order p. Define the Fourier transform of f by the formula: $$\hat f (k) = \sum_{l = 0}^{p-1} f (l) e^{i2\pi kl / p}\qquad(k \in Z_p)$$ Show that if the combined number of zeros of f and $\hat f$ is at least p, then f is identically zero. related: [url]https://artofproblemsolving.com/community/c7h22594[/url]

Evaluate $\int_0^1 (x+3)\sqrt{xe^x}\ dx.$

For each natural number $a$ we denote $\tau (a)$ and $\phi (a)$ the number of natural numbers dividing $a$ and the number of natural numbers less than $a$ that are relatively prime to $a$. Find all natural numbers $n$ for which $n$ has exactly two different prime divisors and $n$ satisfies $\tau (\phi (n))=\phi (\tau (n))$.

Let $I$ be an incenter of $\triangle ABC$. Denote $D, \ S \neq A$ intersections of $AI$ with $BC, \ O(ABC)$ respectively. Let $K, \ L$ be incenters of $\triangle DSB, \ \triangle DCS$. Let $P$ be a reflection of $I$ with the respect to $KL$. Prove that $BP \perp CP$.

Compute the number of positive integers less than or equal to $2015$ that are divisible by $5$ or $13$, but not both.

Let the regular hexagon $ABCDEF$ be inscribed in a circle with center $O$, $N$ be such a point Let $E-N-C$, $M$ a point such that $A- M-C$ and $R$ a point on the circumference, such that $D-N- R$. If $\angle EFR = 90^o$, $\frac{AM}{AC}=\frac{CN}{EC}$ and $AC=\sqrt3$, calculate $AM$. Notation: $A-B-C$ means than points $A,B,C$ are collinear in that order i.e. $ B$ lies between $ A$ and $C$.

Two numbers whose sum is $ 6$ and the absolute value of whose difference is $ 8$ are roots of the equation: $ \textbf{(A)}\ x^2\minus{}6x\plus{}7\equal{}0 \qquad \textbf{(B)}\ x^2\minus{}6x\minus{}7\equal{}0 \qquad \textbf{(C)}\ x^2\plus{}6x\minus{}8\equal{}0 \\ \textbf{(D)}\ x^2\minus{}6x\plus{}8\equal{}0 \qquad \textbf{(E)}\ x^2\plus{}6x\minus{}7\equal{}0$

Compute the sum of the digits of $101^6$.

You use a lock with four dials, each of which is set to a number between 0 and 9 (inclusive). You can never remember your code, so normally you just leave the lock with each dial one higher than the correct value. Unfortunately, last night someone changed all the values to 5. All you remember about your code is that none of the digits are prime, 0, or 1, and that the average value of the digits is 5. How many combinations will you have to try?

Let $ABC$ be a regular triangle. The line passing through the midpoint of $AB$ and parallel to $AC$ meets the minor arc $AB$ of the circumcircle at point $K$. Prove that the ratio $AK:BK$ is equal to the ratio of the side and the diagonal of a regular pentagon.

For constant $k>1$, 2 points $X,\ Y$ move on the part of the first quadrant of the line, which passes through $A(1,\ 0)$ and is perpendicular to the $x$ axis, satisfying $AY=kAX$. Let a circle with radius 1 centered on the origin $O(0,\ 0)$ intersect with line segments $OX,\ OY$ at $P,\ Q$ respectively. Express the maximum area of $\triangle{OPQ}$ in terms of $k$. [i]2011 Tokyo Institute of Technology entrance exam, Problem 3[/i]

A sequence of real numbers $a_1,a_2,\ldots$ satisfies the relation $$a_n=-\max_{i+j=n}(a_i+a_j)\qquad\text{for all}\quad n>2017.$$ Prove that the sequence is bounded, i.e., there is a constant $M$ such that $|a_n|\leq M$ for all positive integers $n$.

Two circles $K_1$ and $K_2$ of different radii intersect at two points $A$ and $B$, let $C$ and $D$ be two points on $K_1$ and $K_2$, respectively, such that $A$ is the midpoint of the segment $CD$. The extension of $DB$ meets $K_1$ at another point $E$, the extension of $CB$ meets $K_2$ at another point $F$. Let $l_1$ and $l_2$ be the perpendicular bisectors of $CD$ and $EF$, respectively. i) Show that $l_1$ and $l_2$ have a unique common point (denoted by $P$). ii) Prove that the lengths of $CA$, $AP$ and $PE$ are the side lengths of a right triangle.

Let $a_n>0$, $n\ge1$. Consider the right triangles $\triangle A_0A_1A_2$, $\triangle A_0A_2A_3,\ldots$, $\triangle A_0A_{n-1}A_n,\ldots,$ as in the figure. (More precisely, for every $n\ge2$ the hypotenuse $A_0A_n$ of $\triangle A_0A_{n-1}A_n$ is a leg of $\triangle A_0A_nA_{n+1}$ with right angle $\angle A_0A_nA_{n+1}$, and the vertices $A_{n-1}$ and $A_{n+1}$ lie on the opposite sides of the straight line $A_0A_n$; also, $|A_{n-1}A_n|=a_n$ for every $n\ge1$.) [img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvYi8yL2M1ZjAxM2I1ZWU0N2E4MzQyYWIzNmQ5OGM3NjJlZjljODdmMTliLnBuZw==&rn=U0VFTU9VUyAyMDEyLnBuZw==[/img] Is it possible for the set of points $\{A_n\mid n\ge0\}$ to be unbounded but the series $\sum_{n=2}^\infty m\angle A_{n-1}A_0A_n$ to be convergent? [i]Note.[/i] A subset $B$ of the plane is bounded if and only if there is a disk $D$ such that $B\subseteq D$.

A function $f:\mathbb{N} \to \mathbb{N}$ is given. If $a,b$ are coprime, then $f(ab)=f(a)f(b)$. Also, if $m,k$ are primes (not necessarily different), then $$f(m+k-3)=f(m)+f(k)-f(3).$$ Find all possible values of $f(11)$.

Given reals $x_1 \ge x_2 \ge ... \ge x_{2003} \ge 0$, show that $$x_1^n - x_2^n + x_2^n - ... - x_{2002}^n + x_{2003}^n \ge (x_1 - x_2 + x_3 - x_4 + ... - x_{2002} + x_{2003})^n$$ for any positive integer $n$.

Prove that for any triangle $ABC$ there exists a point P in the plane of the triangle and three points $A' , B'$ , and $C'$ on the lines $BC, AC$, and $AB$ respectively such that \[AB \cdot PC'= AC \cdot PB'= BC \cdot PA'= 0.3M^2,\] where $M = max\{AB,AC,BC\}$.