Prove that $$\frac{\sin^3 a}{\sin b} +\frac{\cos^3 a}{\cos b} \ge \frac{1}{\cos(a - b)}$$ for all $a$ and $b$ in the interval $(0, \pi/2)$ .

Find the least positive number m such that for any polynimial f(x) with real coefficients, there is a polynimial g(x) with real coefficients (degree not greater than m) such that there exist 2017 distinct number $a_1,a_2,...,a_{2017}$ such that $g(a_i)=f(a_{i+1})$ for i=1,2,...,2017 where indices taken modulo 2017.

Square $ABCD$ has side length $2$. For each $0 \leq r \leq 2$, point $P_r$ is on side $\overline{AB}$ with $AP_r = r$, and square $\Sigma_r$ is constructed with diagonal $\overline{DP_r}$. Let region $\mathcal{R}$ be the set of all points that are in both $\Sigma_0$ and $\Sigma_2$, but not in $\Sigma_r$ for at least one value of $r$. Find the area of the convex hull of $\mathcal{R}$. [i]Proposed by Justin Hsieh[/i]

Does there exist a function $f:\mathbb N\to\mathbb N$ such that $$ f(f(n+1))=f(f(n))+2^{n-1} $$ for any positive integer $n$? (As usual, $\mathbb N$ stands for the set of positive integers.) [i](I. Gorodnin)[/i]

Consider the sequence $$1,7,8,49,50,56,57,343\ldots$$ which consists of sums of distinct powers of$7$, that is, $7^0$, $7^1$, $7^0+7^1$, $7^2$,$\ldots$ in increasing order. At what position will $16856$ occur in this sequence?

In triangle \( ABC \), the incircle is tangent to side \( BC \) at point \( D \), the excircle opposite vertex \( B \) is tangent to line \( AB \) at point \( X \), and the excircle opposite vertex \( C \) is tangent to line \( AC \) at point \( Y \). If \( T \) is the midpoint of segment \( [AD] \) and \( U \) is the circumcenter of triangle \( AXY \), show that \( UT \perp BC \).

Isosceles trapezoid $ABCD$ has side lengths $AB = 6$ and $CD = 12$, while $AD = BC$. It is given that $O$, the circumcenter of $ABCD$, lies in the interior of the trapezoid. The extensions of lines $AD$ and $BC$ intersect at $T$. Given that $OT = 18$, the area of $ABCD$ can be expressed as $a + b\sqrt{c}$ where $a$, $b$, and $c$ are positive integers where $c$ is not divisible by the square of any prime. Compute $a+b+c$. [i]Proposed by Andrew Wen[/i]

Prove that $$\sqrt{x-1}+\sqrt{2x+9}+\sqrt{19-3x}<9$$ for all real $x$ for which the left-hand side is well defined.

Determine for how many positive integers $n\in\{1, 2, \ldots, 2022\}$ it holds that $402$ divides at least one of \[n^2-1, n^3-1, n^4-1\]

Let $ABC$ be an isosceles triangle, $AB = AC$, and let $M$ and $N$ be points on the sides $BC$ and $CA$, respectively, such that $\angle BAM=\angle CNM$. The lines $AB$ and $MN$ meet at $P$. Show that the internal angle bisectors of the angles $BAM$ and $BPM$ meet at a point on the line $BC$.

(a) Positive rational numbers $a, b,$ and $c$ have the property that $\frac{a}{b} + \frac{b}{c} + \frac{c}{a}$ is an integer. Is it possible for $\frac{a}{c} + \frac{c}{b} + \frac{b}{a}$ to also be an integer except for the trivial solution? (b) Positive real numbers $a, b,$ and $c$ have the property that $\frac{a}{b} + \frac{b}{c} + \frac{c}{a}$ is an integer. Is it possible for $\frac{a}{c} + \frac{c}{b} + \frac{b}{a}$ to also be an integer except for the trivial solution? [i](Andrew Brahms, USA)[/i]

Given natural number $a>1$ and different odd prime numbers $p_1,p_2,...,p_n$ with $a^{p_1}\equiv 1$ (mod $p_2$), $a^{p_2}\equiv 1$ (mod $p_3$), ..., $a^{p_n}\equiv 1$(mod $p_1$). Prove that a) $(a-1)\vdots p_i$ for some $i=1,..,n$ b) Can $(a-1)$ be divisible by $p_i $for exactly one $i$ of $i=1,...,n$? I. Bliznets

Let $ ABC$ be an isosceles triangle with $ AB \equal{} AC$. $ M$ is the midpoint of $ BC$ and $ O$ is the point on the line $ AM$ such that $ OB$ is perpendicular to $ AB$. $ Q$ is an arbitrary point on $ BC$ different from $ B$ and $ C$. $ E$ lies on the line $ AB$ and $ F$ lies on the line $ AC$ such that $ E, Q, F$ are distinct and collinear. Prove that $ OQ$ is perpendicular to $ EF$ if and only if $ QE \equal{} QF$.

Assume that the earth is a perfect sphere. A plane flies between $30^o N$ $45^o W$ and $30^o N$ $45^o E$ along the shortest possible route. Let $\theta$ be the northernmost latitude that the plane flies over. Compute $\sin \theta$.

Determine the smallest value of $(a+5)^2+(b-2)^2+(c-9)^2$ for all real numbers $a, b, c$ satisfying $a^2+b^2+c^2-ab-bc-ca=3$.

Calculate the following indefinite integrals. [1] $\int \frac{(x^2-1)^2}{x^4}dx$ [2] $\int \frac{e^{3x}}{\sqrt{e^x+1}}dx$ [3] $\int \sin 2x\cos 3xdx$ [4] $\int x\ln (x+1)dx$ [5] $\int \frac{x}{(x+3)^2}dx$

A set $M$ of elements $u, v, w$ is called a semigroup if an operation is defined in it is which uniquely assigns an element $w$ from $M$ to every ordered pair $(u, v)$ of elements from $M$ (you write $u \otimes v = w$) and if this algebraic operation is associative, i.e. if for all elements $u, v,w$ from $M$: $$(u \otimes v) \otimes w = u \otimes (v \otimes w).$$ Now let $c$ be a positive real number and let $M$ be the set of all non-negative real numbers that are smaller than $c$. For each two numbers $u, v$ from $M$ we define: $$u \otimes v = \dfrac{u + v}{1 + \dfrac{uv}{c^2}}$$ Investigate a) whether $M$ is a semigroup; b) whether this semigroup is regular, i.e. whether from $u \otimes v_1 = u\otimes v_2$ always $v_1 = v_2$ and from $v_1 \otimes u = v_2 \otimes u$ also $v_1 = v_2$ follows.

Let $ABCD$ be an isosceles trapezoid, whose dimensions are $AB = 6$, $BC=5=DA$, and $CD=4$. Draw circles of radius 3 centered at $A$ and $B$, and circles of radius 2 centered at $C$ and $D$. A circle contained within the trapezoid is tangent to all four of these circles. Its radius is $\frac{-k+m\sqrt{n}}p$, where $k$, $m$, $n$, and $p$ are positive integers, $n$ is not divisible by the square of any prime, and $k$ and $p$ are relatively prime. Find $k+m+n+p$.

The points of a circle are colored in green and yellow, such that every equilateral triangle inscribed in the circle has exactly 2 vertices colored in yellow. Prove that there exist a square inscribed in the circle which has at least 3 vertices colored in yellow.

A plane passing through the center of a cube intersects the cube in a cyclic hexagon. Show that this hexagon is regular.

The incircles of the faces $ABC$ and $ABD$ of a tetrahedron $ABCD$ are tangent to the edge $AB$ in the same point. Prove that the points of tangency of these incircles to the edges $AC,BC,AD,BD$ are concyclic.

Four boys bought a boat for $\textdollar 60$. The first boy paid one half of the sum of the amounts paid by the other boys; the second boy paid one third of the sum of the amounts paid by the other boys; and the third boy paid one fourth of the sum of the amounts paid by the other boys. How much did the fourth boy pay? $\textbf{(A) }\textdollar 10\qquad\textbf{(B) }\textdollar 12\qquad\textbf{(C) }\textdollar 13\qquad\textbf{(D) }\textdollar 14\qquad \textbf{(E) }\textdollar 15$

Let $ABC$ be an non-isosceles triangle with incenter $I$, circumcenter $O$ and a point $D$ on segment $BC $such that $(BID) $cut segments $AB $ at$ E $and $(CID) $cuts segment $AC $at $F$ Circle $(DEF)$ cuts segments $AB$,$AC $again at $M,N$. Let $P$ The intersection of $IB$ and $DE $ , $Q$ The intersection of $IC$and $DF$ . Prove that $EN,FM,PQ $are parallel and the median of vertex $I$in triangle $IPQ$ bisects the arc $BAC$ of $(O)$.

On day $1$ of the new year, John Adams and Samuel Adams each drink one gallon of tea. For each positive integer $n$, on the $n$th day of the year, John drinks $n$ gallons of tea and Samuel drinks $n^2$ gallons of tea. After how many days does the combined tea intake of John and Samuel that year first exceed $900$ gallons? [i]Proposed by Aidan Duncan[/i] [hide=Solution] [i]Solution. [/i] $\boxed{13}$ The total amount that John and Samuel have drank by day $n$ is $$\dfrac{n(n+1)(2n+1)}{6}+\dfrac{n(n+1)}{2}=\dfrac{n(n+1)(n+2)}{3}.$$ Now, note that ourdesired number of days should be a bit below $\sqrt[3]{2700}$. Testing a few values gives $\boxed{13}$ as our answer. [/hide]

Let $n$ be a positive integer. Each of the numbers $1,2,3,\ldots,100$ is painted with one of $n$ colors in such a way that two distinct numbers with a sum divisible by $4$ are painted with different colors. Determine the smallest value of $n$ for which such a situation is possible.