Let $\phi(n)$ denote the number of positive integers less than or equal to $n$ which are relatively prime to $n$. Compute $\displaystyle \sum_{i=1}^{\phi(2023)} \dfrac{\gcd(i,\phi(2023))}{\phi(2023)}$. [i]Proposed by Giacomo Rizzo[/i]

Find the functions $f: \mathbb{R} \to (0, \infty)$ which satisfy $$2^{-x-y} \le \frac{f(x)f(y)}{(x^2+1)(y^2+1)} \le \frac{f(x+y)}{(x+y)^2+1},$$ for all $x,y \in \mathbb{R}.$

We are given a trapezoid $ABCD$ with $AB \parallel CD$, $CD=2AB$ and $DB \perp BC$. Let $E$ be the intersection of lines $DA$ and $CB$, and $M$ be the midpoint of $DC$. (a) Prove that $ABMD$ is a rhombus. (b) Prove that triangle $CDE$ is isosceles. (c) If $AM$ and $BD$ meet at $O$, and $OE$ and $AB$ meet at $N,$ prove that the line $DN$ bisects segment $EB$.

Let $r$ and $k$ be positive integers such that all prime divisors of $r$ are greater than $50$. A positive integer, whose decimal representation (without leading zeroes) has at least $k$ digits, will be called [i]nice[/i] if every sequence of $k$ consecutive digits of this decimal representation forms a number (possibly with leading zeroes) which is a multiple of $r$. Prove that if there exist infinitely many nice numbers, then the number $10^k-1$ is nice as well.

In parallelogram $ABCD$, $AC=10$ and $BD=28$. The points $K$ and $L$ in the plane of $ABCD$ move in such a way that $AK=BD$ and $BL=AC$. Let $M$ and $N$ be the midpoints of $CK$ and $DL$, respectively. What is the maximum walue of $\cot^2 (\tfrac{\angle BMD}{2})+\tan^2(\tfrac{\angle ANC}{2})$ ?

If two points are selected at random on a fixed circle and the chord between the two points is drawn, what is the probability that its length exceeds the radius of the circle?

Let $n =123456789101112 ... 998999$ be the natural number where is obtained by writing the natural numbers from $1$ to $999$ one after the other. What is the $1997$-th digit number in $n$?

Elmo calls a monic polynomial with real coefficients [i]tasty[/i] if all of its coefficients are in the range $[-1,1]$. A monic polynomial $P$ with real coefficients and complex roots $\chi_1,\cdots,\chi_m$ (counted with multiplicity) is given to Elmo, and he discovers that there does not exist a monic polynomial $Q$ with real coefficients such that $PQ$ is tasty. Find all possible values of $\max\left(|\chi_1|,\cdots,|\chi_m|\right)$. [i]Proposed by Carl Schildkraut[/i]

The figure shown can be folded into the shape of a cube. In the resulting cube, which of the lettered faces is opposite the face marked $x$? [asy] defaultpen(linewidth(0.7)); path p=origin--(0,1)--(1,1)--(1,2)--(2,2)--(2,3); draw(p^^(2,3)--(4,3)^^shift(2,0)*p^^(2,0)--origin); draw(shift(1,0)*p, dashed); label("$x$", (0.3,0.5), E); label("$A$", (1.3,0.5), E); label("$B$", (1.3,1.5), E); label("$C$", (2.3,1.5), E); label("$D$", (2.3,2.5), E); label("$E$", (3.3,2.5), E);[/asy] $ \mathbf{(A)}\; A\qquad \mathbf{(B)}\; B\qquad \mathbf{(C)}\; C\qquad \mathbf{(D)}\; D\qquad \mathbf{(E)}\; E$

In a triangle $ABC$, let $AB$ be the shortest side. Points $X$ and $Y$ are given on the circumcircle of $\triangle ABC$ such that $CX=AX+BX$ and $CY=AY+BY$. Prove that $\measuredangle XCY<60^{o}$.

Find the range of the function $f(x)=\frac{\sqrt{x^2+1}}{x-1}$.

Let $C$ be a continuous closed curve in the plane which does not cross itself and let $Q$ be a point inside $C$. Show that there exists points $P_1$ and $P_2$ on $C$ such that $Q$ is the midpoint of the line segment $P_1P_2.$

Determine the value of $a,b$ for which $\int_0^1 (\sqrt{1-x}-ax-b)^2 dx$ is minimized.

Aaliyah rolls two standard 6-sided dice. She notices that the product of the two numbers rolled is a multiple of 6. Which of the following integers [i]cannot[/i] be the sum of the two numbers? $\textbf{(A) } 5\qquad\textbf{(B) } 6\qquad\textbf{(C) } 7\qquad\textbf{(D) } 8\qquad\textbf{(E) } 9$

Let be three nonnegative integers $ m,n,p $ and three real numbers $ x,y,z $ such that $ 2^mx+2^ny+2^pz\ge 0. $ Prove: $$ 2^m\left( 2^x-1 \right)+2^n\left( 2^y-1 \right)+2^p\left( 2^z-1 \right)\ge 0 $$ [i]Cristinel Mortici[/i]

Positive real numbers $x,y,z$ have the sum $1$. Prove that $\sqrt{7x+3}+ \sqrt{7y+3}+\sqrt{7z+3} \le 7$. Can number $7$ on the right hand side be replaced with a smaller constant?

Three lines $k, l, m$ are drawn through a point $P$ inside a triangle $ABC$ such that $k$ meets $AB$ at $A_1$ and $AC$ at $A_2 \ne A_1$ and $PA_1 = PA_2$, $l $ meets $BC$ at $B_1$ and $BA$ at $B_2 \ne B_1$ and $PB_1 = PB_2$, $m$ meets $CA$ at $C_1$ and $CB$ at $C_2\ne C_1$ and $PC_1=PC_2$. Prove that the lines $k,l,m$ are uniquely determined by these conditions. Find point $P$ for which the triangles $AA_1A_2, BB_1B_2, CC_1C_2$ have the same area and show that this point is unique.

The product $N$ of three positive integers is $6$ times their sum, and one of the integers is the sum of the other two. Find the sum of all possible values of $N.$

Let $a,b$, and $x$ be positive integers such that $x^{a+b}=a^b{b}$. Prove that $a=x$ and $b=x^{x}$.

For which $m > 1$ is $(m -1)!$ divisible by $m$?

A circle with a circumscribed and an inscribed square centered at the origin $ O$ of a rectangular coordinate system with positive $ x$ and $ y$ axes $ OX$ and $ OY$ is shown in each figure $ I$ to $ IV$ below. [asy] size((400)); draw((0,0)--(22,0), EndArrow); draw((10,-10)--(10,12), EndArrow); draw((25,0)--(47,0), EndArrow); draw((35,-10)--(35,12), EndArrow); draw((-25,0)--(-3,0), EndArrow); draw((-15,-10)--(-15,12), EndArrow); draw((-50,0)--(-28,0), EndArrow); draw((-40,-10)--(-40,12), EndArrow); draw(Circle((-40,0),6)); draw(Circle((-15,0),6)); draw(Circle((10,0),6)); draw(Circle((35,0),6)); draw((-34,0)--(-40,6)--(-46,0)--(-40,-6)--(-34,0)--(-34,6)--(-46,6)--(-46,-6)--(-34,-6)--cycle); draw((-6.5,0)--(-15,8.5)--(-23.5,0)--(-15,-8.5)--cycle); draw((-10.8,4.2)--(-19.2,4.2)--(-19.2,-4.2)--(-10.8,-4.2)--cycle); draw((14.2,4.2)--(5.8,4.2)--(5.8,-4.2)--(14.2,-4.2)--cycle); draw((16,6)--(4,6)--(4,-6)--(16,-6)--cycle); draw((41,0)--(35,6)--(29,0)--(35,-6)--cycle); draw((43.5,0)--(35,8.5)--(26.5,0)--(35,-8.5)--cycle); label("I", (-49,9)); label("II", (-24,9)); label("III", (1,9)); label("IV", (26,9)); label("X", (-28,0), S); label("X", (-3,0), S); label("X", (22,0), S); label("X", (47,0), S); label("Y", (-40,12), E); label("Y", (-15,12), E); label("Y", (10,12), E); label("Y", (35,12), E);[/asy] The inequalities \[ |x| \plus{} |y| \leq \sqrt {2(x^2 \plus{} y^2)} \leq 2\mbox{Max}(|x|, |y|)\] are represented geometrically* by the figure numbered $ \textbf{(A)}\ I \qquad \textbf{(B)}\ II \qquad \textbf{(C)}\ III \qquad \textbf{(D)}\ IV \qquad \textbf{(E)}\ \mbox{none of these}$ *An inequality of the form $ f(x, y) \leq g(x, y)$, for all $ x$ and $ y$ is represented geometrically by a figure showing the containment \[ \{\mbox{The set of points }(x, y)\mbox{ such that }g(x, y) \leq a\} \subset\\ \{\mbox{The set of points }(x, y)\mbox{ such that }f(x, y) \leq a\}\] for a typical real number $ a$.

The points $A, B, C, D, P$ lie on an circle as shown in the figure such that $\angle AP B = \angle BPC = \angle CPD$. Prove that the lengths of the segments are denoted by $a, b, c, d$ by $\frac{a + c}{b + d} =\frac{b}{c}$. [img]https://cdn.artofproblemsolving.com/attachments/a/2/ba8965f5d7d180426db26e8f7dd5c7ad02c440.png[/img]

Find all pairs $(x,y)$ of positive integers that satisfy the equation \[y^{2}=x^{3}+16.\]

Find all functions $f : R \to R$ such that $x[f(x + y) - f (x - y)] = 4y f (x)$ for any real numbers $x, y$.

Find all triplets $(x,y,p)$ of positive integers such that $p$ be a prime number and $\frac{xy^3}{x+y}=p$