For all pairwise distinct positive real numbers $a, b, c$ such that $abc = 1$, prove that $$\frac{\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+1}{(a+b+c+1)^2}+\frac{3}{8}\sqrt[3]{\frac{(a^2-b^2)^3+(b^2-c^2)^3+(c^2-a^2)^3}{(a-b)^3+(b-c)^3+(c-a)^3}}\geq 1.$$

Each term of the sequence $5, 12, 19, 26, \cdots$ is $7$ more than the term that precedes it. What is the first term of the sequence that is greater than $2017$? $\text{(A) }2018\qquad\text{(B) }2019\qquad\text{(C) }2020\qquad\text{(D) }2021\qquad\text{(E) }2022$

Let $a,b,c$ be positive rational numbers such that $\sqrt a+\sqrt b=c$. Show that $\sqrt a$ and $\sqrt b$ are also rational.

In triangle $ABC$ on side $AC$ are points $K$, $L$ and $M$ such that $BK$ is an angle bisector of $\angle ABL$, $BL$ is an angle bisector of $\angle KBM$ and $BM$ is an angle bisector of $\angle LBC$, respectively. Prove that $4 \cdot LM <AC$ and $3\cdot \angle BAC - \angle ACB < 180^{\circ}$

Consider the function $ f(\theta) \equal{} \int_0^1 |\sqrt {1 \minus{} x^2} \minus{} \sin \theta|dx$ in the interval of $ 0\leq \theta \leq \frac {\pi}{2}$. (1) Find the maximum and minimum values of $ f(\theta)$. (2) Evaluate $ \int_0^{\frac {\pi}{2}} f(\theta)\ d\theta$.

Inside square $ABCD$ consider a point $M$. Prove that the points of intersection of the medians of triangles $ABM, BCM, CDM$ and $DAM$ form a square. (V Prasolov)

Let $ABCD$ be a convex quadrilateral with $\angle B= \angle D$. Prove that the midpoint of $BD$ lies on the common internal tangent to the incircles of triangles $ABC$ and $ACD$.

Let $P(x)$ be a quadratic polynomial satisfying the following conditions: [list] [*] $P(x)$ has leading coefficient $1$. [*] $P(x)$ has nonnegative integer roots that are at most $2022$. [*] the set of the roots of $P(x)$ is a subset of the set of the roots of $P(P(x))$. [/list] Let $S$ be the set of all such possible $P(x)$, and let $Q(x)$ be the polynomial obtained upon summing all the elements of $S$. Find the sum of the roots of $Q(x)$.

Define $$P(x) =(x-1^2)(x-2^2)\cdots(x-100^2).$$ How many integers $n$ are there such that $P(n)\leq 0$? $\textbf{(A) } 4900 \qquad \textbf{(B) } 4950\qquad \textbf{(C) } 5000\qquad \textbf{(D) } 5050 \qquad \textbf{(E) } 5100$

Find the smallest odd natural number $N$ such that $N^2$ is the sum of an odd number (greater than $1$) of squares of adjacent positive integers.

Two circles with radius $2$, $\omega_1$ and $\omega_2$, are centered at $O_1$ and $O_2$ respectively. The circles $\omega_1$ and $\omega_2$ are externally tangent to each other and internally tangent to a larger circle $\omega$ centered at $O$ at points $A$ and $B$, respectively. Let $M$ be the midpoint of minor arc $AB$. Let $P$ be the intersection of $\omega_1$ and $O_1M$, and let $Q$ be the intersection of $\omega_2$ and $O_2M$. Given that there is a point $R$ on $\omega$ such that $\triangle PQR$ is equilateral, the radius of $\omega$ can be written as $\frac{a+\sqrt{b}}{c}$ where $a$, $b$, and $c$ are positive integers and $a$ and $c$ are relatively prime. Find $a+b+c$.

We say that a set $S$ of integers is [i]rootiful[/i] if, for any positive integer $n$ and any $a_0, a_1, \cdots, a_n \in S$, all integer roots of the polynomial $a_0+a_1x+\cdots+a_nx^n$ are also in $S$. Find all rootiful sets of integers that contain all numbers of the form $2^a - 2^b$ for positive integers $a$ and $b$.

For which $k$ does the system $x^2 - y^2 = 0, (x-k)^2 + y^2 = 1$ have exactly: (i) two, (ii) three real solutions?

Determine the largest positive integer $k$ for which the following statement is true: given $k$ distinct subsets of the set $\{1, 2, 3, \dots , 2023\}$, each with $1011$ elements, it is possible partition the subsets into two collections so that any two subsets in one same collection have some element in common.

How many numbers among $1,2,\ldots,2012$ have a positive divisor that is a cube other than $1$? $\text{(A) }346\qquad\text{(B) }336\qquad\text{(C) }347\qquad\text{(D) }251\qquad\text{(E) }393$

Given six three-element subsets of a finite set $X$, show that it is possible to color the elements of $X$ in two colors so that none of the given subsets is in one color

Five dice are rolled. The product of the faces are then computed. Which result has a larger probability of occurring; $180$ or $144$?

A customer who intends to purchase an appliance has three coupons, only one of which may be used: Coupon 1: $10\%$ off the listed price if the listed price is at least $\$50$ Coupon 2: $\$20$ off the listed price if the listed price is at least $\$100$ Coupon 3: $18\%$ off the amount by which the listed price exceeds $\$100$ For which of the following listed prices will coupon $1$ offer a greater price reduction than either coupon $2$ or coupon $3$? $\textbf{(A) }\$179.95\qquad \textbf{(B) }\$199.95\qquad \textbf{(C) }\$219.95\qquad \textbf{(D) }\$239.95\qquad \textbf{(E) }\$259.95\qquad$

Each half of this figure is composed of 3 red triangles, 5 blue triangles and 8 white triangles. When the upper half is folded down over the centerline, 2 pairs of red triangles coincide, as do 3 pairs of blue triangles. There are 2 red-white pairs. How many white pairs coincide? [asy] draw((0,0)--(4,4*sqrt(3))); draw((1,-sqrt(3))--(5,3*sqrt(3))); draw((2,-2*sqrt(3))--(6,2*sqrt(3))); draw((3,-3*sqrt(3))--(7,sqrt(3))); draw((4,-4*sqrt(3))--(8,0)); draw((8,0)--(4,4*sqrt(3))); draw((7,-sqrt(3))--(3,3*sqrt(3))); draw((6,-2*sqrt(3))--(2,2*sqrt(3))); draw((5,-3*sqrt(3))--(1,sqrt(3))); draw((4,-4*sqrt(3))--(0,0)); draw((3,3*sqrt(3))--(5,3*sqrt(3))); draw((2,2*sqrt(3))--(6,2*sqrt(3))); draw((1,sqrt(3))--(7,sqrt(3))); draw((-1,0)--(9,0)); draw((1,-sqrt(3))--(7,-sqrt(3))); draw((2,-2*sqrt(3))--(6,-2*sqrt(3))); draw((3,-3*sqrt(3))--(5,-3*sqrt(3)));[/asy] $ \text{(A)}\ 4\qquad\text{(B)}\ 5\qquad\text{(C)}\ 6\qquad\text{(D)}\ 7\qquad\text{(E)}\ 9 $

Let $X=\mathbb{Z}^3$ denote the set of all triples $(a,b,c)$ of integers. Define $f: X \to X$ by \[ f(a,b,c) = (a+b+c, ab+bc+ca, abc) . \] Find all triples $(a,b,c)$ such that \[ f(f(a,b,c)) = (a,b,c) . \]

Fold the next seven corners into a rectangle. [img]https://cdn.artofproblemsolving.com/attachments/b/b/2b8b9d6d4b72024996a66d41f865afb91bb9b7.png[/img]

If Xiluva puts two oranges in each basket, four oranges are in excess. If she puts five oranges in each basket, one basket is in excess. How many oranges and baskets has Xiluva?

Let $ABC$ be a triangle with $AB<AC$ and incenter $I.$ A point $D$ lies on segment $AC$ such that $AB=AD,$ and the line $BI$ intersects $AC$ at $E.$ Suppose the line $CI$ intersects $BD$ at $F,$ and $G$ lies on segment $DI$ such that $FD=FG.$ Prove that the lines $AG$ and $EF$ intersect on the circumcircle of triangle $CEI.$ \\ Proposed by Avan Lim Zenn Ee, Malaysia

Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $$f(x^2y)=f(xy)+yf(f(x)+y)$$ for all real numbers $x$ and $y$.

Find the number of ordered triples of positive integers $(a, b, c)$ such that \[6a + 10b + 15c = 3000.\]