Positive integers $a, b, c, d$ satisfy the inequality $ad - bc > 1$. Prove that at least one of the numbers $a, b, c, d$ is not divisible by $ad - bc$. (A Spivak)

Maybe well known: $p$ a prime number, $n$ an integer. Prove that $n$ divides $\phi(p^n-1)$ where $\phi(x)$ is the Euler function.

In a cyclic quadrilateral $ABCD$ with $AB=AD$ points $M$,$N$ lie on the sides $BC$ and $CD$ respectively so that $MN=BM+DN$ . Lines $AM$ and $AN$ meet the circumcircle of $ABCD$ again at points $P$ and $Q$ respectively. Prove that the orthocenter of the triangle $APQ$ lies on the segment $MN$ .

Given the positive real numbers $a_{1},a_{2},\dots,a_{n},$ such that $n>2$ and $a_{1}+a_{2}+\dots+a_{n}=1,$ prove that the inequality \[ \frac{a_{2}\cdot a_{3}\cdot\dots\cdot a_{n}}{a_{1}+n-2}+\frac{a_{1}\cdot a_{3}\cdot\dots\cdot a_{n}}{a_{2}+n-2}+\dots+\frac{a_{1}\cdot a_{2}\cdot\dots\cdot a_{n-1}}{a_{n}+n-2}\leq\frac{1}{\left(n-1\right)^{2}}\] does holds.

Find all positive integers $n$, for which there exists a positive integer $k$ such that for every positive divisor $d$ of $n$, the number $d - k$ is also a (not necessarily positive) divisor of $n$.

Find all the natural $n$ such that $n!$ is not divisible by $n^2$.

Three nonnegative real numbers $ r_1$, $ r_2$, $ r_3$ are written on a blackboard. These numbers have the property that there exist integers $ a_1$, $ a_2$, $ a_3$, not all zero, satisfying $ a_1r_1 \plus{} a_2r_2 \plus{} a_3r_3 \equal{} 0$. We are permitted to perform the following operation: find two numbers $ x$, $ y$ on the blackboard with $ x \le y$, then erase $ y$ and write $ y \minus{} x$ in its place. Prove that after a finite number of such operations, we can end up with at least one $ 0$ on the blackboard.

Let the isosceles right triangle $ABC$ with $\angle A= 90^o$ . The points $E$ and $F$ are taken on the ray $AC$ so that $\angle ABE = 15^o$ and $CE = CF$. Determine the measure of the angle $CBF$.

Let $\mathbb{R}$ be the set of real numbers. Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a function such that \[f(x+y)f(x-y)\geqslant f(x)^2-f(y)^2\] for every $x,y\in\mathbb{R}$. Assume that the inequality is strict for some $x_0,y_0\in\mathbb{R}$. Prove that either $f(x)\geqslant 0$ for every $x\in\mathbb{R}$ or $f(x)\leqslant 0$ for every $x\in\mathbb{R}$.

If three integers $p, q$ and $r$ apply that $$p + q^2 = r ^2.$$Show that $6$ adds up to $pqr$ .

Suppose $A,B\in M_3(\mathbb C)$, $B\ne0$, and $AB=0$. Prove that there exists $D\in M_3(\mathbb C)$ with $D\ne0$ such that $AD=DA=0$.

Find all triples $(f,g,h)$ of injective functions from the set of real numbers to itself satisfying \begin{align*} f(x+f(y)) &= g(x) + h(y) \\ g(x+g(y)) &= h(x) + f(y) \\ h(x+h(y)) &= f(x) + g(y) \end{align*} for all real numbers $x$ and $y$. (We say a function $F$ is [i]injective[/i] if $F(a)\neq F(b)$ for any distinct real numbers $a$ and $b$.) [i]Proposed by Evan Chen[/i]

Let $ABC$ be a triangle whose side lengths are, as usual, denoted by $a=|BC|,$ $b=|CA|,$ $c=|AB|.$ Denote by $m_a,m_b,m_c$, respectively, the lengths of the medians which connect $A,B,C$, respectively, with the centers of the corresponding opposite sides. (a) Prove that $2m_a<b+c$. Deduce that $m_a+m_b+m_c<a+b+c$. (b) Give an example of (i) a triangle in which $m_a>\sqrt{bc}$; (ii) a triangle in which $m_a\le \sqrt{bc}$.

Let $ a$ and $ b$ be natural numbers such that \[ \frac{a}{b}=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\cdots-\frac{1}{1318}+\frac{1}{1319}. \] Prove that $ a$ is divisible by $ 1979$.

Each of Marta's fish has exactly $3$ fins or $4$ fins. Marta's $13$ fish have a total of $44$ fins. How many $3$-finned fish does Marta have? $\text{(A) }5\qquad\text{(B) }6\qquad\text{(C) }7\qquad\text{(D) }8\qquad\text{(E) }9$

What is the difference between the sum of the first $ 2003$ even counting numbers and the sum of the first $ 2003$ odd counting numbers? $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 2003 \qquad \textbf{(E)}\ 4006$

Let $H$ be the orthocenter of the triangle $ABC$. Let $M$ and $N$ be the midpoints of the sides $AB$ and $AC$, respectively. Assume that $H$ lies inside the quadrilateral $BMNC$ and that the circumcircles of triangles $BMH$ and $CNH$ are tangent to each other. The line through $H$ parallel to $BC$ intersects the circumcircles of the triangles $BMH$ and $CNH$ in the points $K$ and $L$, respectively. Let $F$ be the intersection point of $MK$ and $NL$ and let $J$ be the incenter of triangle $MHN$. Prove that $F J = F A$.

[b]p1.[/b] David runs at $3$ times the speed of Alice. If Alice runs $2$ miles in $30$ minutes, determine how many minutes it takes for David to run a mile. [b]p2.[/b] Al has $2019$ red jelly beans. Bob has $2018$ green jelly beans. Carl has $x$ blue jelly beans. The minimum number of jelly beans that must be drawn in order to guarantee $2$ jelly beans of each color is $4041$. Compute $x$. [b]p3.[/b] Find the $7$-digit palindrome which is divisible by $7$ and whose first three digits are all $2$. [b]p4.[/b] Determine the number of ways to put $5$ indistinguishable balls in $6$ distinguishable boxes. [b]p5.[/b] A certain reduced fraction $\frac{a}{b}$ (with $a,b > 1$) has the property that when $2$ is subtracted from the numerator and added to the denominator, the resulting fraction has $\frac16$ of its original value. Find this fraction. [b]p6.[/b] Find the smallest positive integer $n$ such that $|\tau(n +1)-\tau(n)| = 7$. Here, $\tau(n)$ denotes the number of divisors of $n$. [b]p7.[/b] Let $\vartriangle ABC$ be the triangle such that $AB = 3$, $AC = 6$ and $\angle BAC = 120^o$. Let $D$ be the point on $BC$ such that $AD$ bisect $\angle BAC$. Compute the length of $AD$. [b]p8.[/b] $26$ points are evenly spaced around a circle and are labeled $A$ through $Z$ in alphabetical order. Triangle $\vartriangle LMT$ is drawn. Three more points, each distinct from $L, M$, and $T$ , are chosen to form a second triangle. Compute the probability that the two triangles do not overlap. [b]p9.[/b] Given the three equations $a +b +c = 0$ $a^2 +b^2 +c^2 = 2$ $a^3 +b^3 +c^3 = 19$ find $abc$. [b]p10.[/b] Circle $\omega$ is inscribed in convex quadrilateral $ABCD$ and tangent to $AB$ and $CD$ at $P$ and $Q$, respectively. Given that $AP = 175$, $BP = 147$,$CQ = 75$, and $AB \parallel CD$, find the length of $DQ$. [b]p11. [/b]Let $p$ be a prime and m be a positive integer such that $157p = m^4 +2m^3 +m^2 +3$. Find the ordered pair $(p,m)$. [b]p12.[/b] Find the number of possible functions $f : \{-2,-1, 0, 1, 2\} \to \{-2,-1, 0, 1, 2\}$ that satisfy the following conditions. (1) $f (x) \ne f (y)$ when $x \ne y$ (2) There exists some $x$ such that $f (x)^2 = x^2$ [b]p13.[/b] Let $p$ be a prime number such that there exists positive integer $n$ such that $41pn -42p^2 = n^3$. Find the sum of all possible values of $p$. [b]p14.[/b] An equilateral triangle with side length $ 1$ is rotated $60$ degrees around its center. Compute the area of the region swept out by the interior of the triangle. [b]p15.[/b] Let $\sigma (n)$ denote the number of positive integer divisors of $n$. Find the sum of all $n$ that satisfy the equation $\sigma (n) =\frac{n}{3}$. [b]p16[/b]. Let $C$ be the set of points $\{a,b,c\} \in Z$ for $0 \le a,b,c \le 10$. Alice starts at $(0,0,0)$. Every second she randomly moves to one of the other points in $C$ that is on one of the lines parallel to the $x, y$, and $z$ axes through the point she is currently at, each point with equal probability. Determine the expected number of seconds it will take her to reach $(10,10,10)$. [b]p17.[/b] Find the maximum possible value of $$abc \left( \frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)^3$$ where $a,b,c$ are real such that $a +b +c = 0$. [b]p18.[/b] Circle $\omega$ with radius $6$ is inscribed within quadrilateral $ABCD$. $\omega$ is tangent to $AB$, $BC$, $CD$, and $DA$ at $E, F, G$, and $H$ respectively. If $AE = 3$, $BF = 4$ and $CG = 5$, find the length of $DH$. [b]p19.[/b] Find the maximum integer $p$ less than $1000$ for which there exists a positive integer $q$ such that the cubic equation $$x^3 - px^2 + q x -(p^2 -4q +4) = 0$$ has three roots which are all positive integers. [b]p20.[/b] Let $\vartriangle ABC$ be the triangle such that $\angle ABC = 60^o$,$\angle ACB = 20^o$. Let $P$ be the point such that $CP$ bisects $\angle ACB$ and $\angle PAC = 30^o$. Find $\angle PBC$. PS. You had better use hide for answers.

Let $p$ be a prime number and let $m$ and $n$ be positive integers with $p^2 + m^2 = n^2$. Prove that $m> p$. (Karl Czakler)

Let $ABCD$ be a convex quadrilateral with $AB=BD=DC$ and $AB \perp BD \perp DC$. Let $M$ be the midpoint of segment $BC$. Show that $\angle BAM+\angle DCA=45^\circ$.

In a convex pentagon $ABCDE$ the following conditions hold : $AB \parallel CD$, $BC \parallel DE$, and $\angle BAE = \angle AED$. Prove that $AB + BC = CD + DE$ [i]Proposed by Anton Trygub[/i]

The letters $P$, $Q$, $R$, $S$, and $T$ represent numbers located on the number line as shown. [asy] unitsize(36); draw((-4,0)--(4,0)); draw((-3.9,0.1)--(-4,0)--(-3.9,-0.1)); draw((3.9,0.1)--(4,0)--(3.9,-0.1)); for (int i = -3; i <= 3; ++i) { draw((i,-0.1)--(i,0)); } label("$-3$",(-3,-0.1),S); label("$-2$",(-2,-0.1),S); label("$-1$",(-1,-0.1),S); label("$0$",(0,-0.1),S); label("$1$",(1,-0.1),S); label("$2$",(2,-0.1),S); label("$3$",(3,-0.1),S); draw((-3.7,0.1)--(-3.6,0)--(-3.5,0.1)); draw((-3.6,0)--(-3.6,0.25)); label("$P$",(-3.6,0.25),N); draw((-1.3,0.1)--(-1.2,0)--(-1.1,0.1)); draw((-1.2,0)--(-1.2,0.25)); label("$Q$",(-1.2,0.25),N); draw((0.1,0.1)--(0.2,0)--(0.3,0.1)); draw((0.2,0)--(0.2,0.25)); label("$R$",(0.2,0.25),N); draw((0.8,0.1)--(0.9,0)--(1,0.1)); draw((0.9,0)--(0.9,0.25)); label("$S$",(0.9,0.25),N); draw((1.4,0.1)--(1.5,0)--(1.6,0.1)); draw((1.5,0)--(1.5,0.25)); label("$T$",(1.5,0.25),N); [/asy] Which of the following expressions represents a negative number? $\text{(A)}\ P-Q \qquad \text{(B)}\ P\cdot Q \qquad \text{(C)}\ \dfrac{S}{Q}\cdot P \qquad \text{(D)}\ \dfrac{R}{P\cdot Q} \qquad \text{(E)}\ \dfrac{S+T}{R}$

Given is a convex octagon $A_1A_2 \ldots A_8$. Given a triangulation $T$, one can take two triangles $\triangle A_iA_jA_k$ and $\triangle A_iA_kA_l$ and replace them with $\triangle A_iA_jA_l$ and $\triangle A_jA_lA_k$. Find the minimal number of operations $k$ we have to do so that for any pair of triangulations $T_1, T_2$, we can reach $T_2$ from $T_1$ using at most $k$ operations.

All twelve points on the circle are at equal distances. The only marked point inside is the center of the circle. Determine which part of the whole circle in the picture is filled in. [img]https://cdn.artofproblemsolving.com/attachments/9/0/9a6af9cef6a4bb03fb4d3eef715f3fd77c74b3.png[/img]

Let $n{}$ be a positive integer. Show that there exists a polynomial $f{}$ of degree $n{}$ with integral coefficients such that \[f^2=(x^2-1)g^2+1,\] where $g{}$ is a polynomial with integral coefficients.