Show that $\frac{1}{\pi} \arccos \left( \frac{1}{\sqrt{2003}} \right)$ is irrational.

The sum of positive numbers $a, b, c$ is equal to $\pi/2$. Prove that $$\cos a + \cos b + \cos c > \sin a + \sin b + \sin c.$$

Let $n$ be a positive integer. If the equation $2x+2y+z=n$ has $28$ solutions in positive integers $x, y$ and $z$, then $n$ must be either $ \textbf{(A)}\ 14\ \text{or}\ 15 \qquad\textbf{(B)}\ 15\ \text{or}\ 16 \qquad\textbf{(C)}\ 16\ \text{or}\ 17 \qquad\textbf{(D)}\ 17\ \text{or}\ 18 \qquad\textbf{(E)}\ 18\ \text{or}\ 19 $

For a positive integer $n$, an $n$-configuration is a family of sets $\left\langle A_{i,j}\right\rangle_{1\le i,j\le n}.$ An $n$-configuration is called [i]sweet[/i] if for every pair of indices $(i, j)$ with $1\le i\le n -1$ and $1\le j\le n$ we have $A_{i,j}\subseteq A_{i+1,j}$ and $A_{j,i}\subseteq A_{j,i+1}.$ Let $f(n, k)$ denote the number of sweet $n$-configurations such that $A_{n,n}\subseteq \{1, 2,\ldots , k\}$. Determine which number is larger: $f\left(2024, 2024^2\right)$ or $f\left(2024^2, 2024\right).$

A convex polygon with $2n$ sides is called [i]rhombic [/i] if its sides are equal and all pairs of opposite sides are parallel. A rhombic polygon can be partitioned into rhombic quadrilaterals. For what value of$ n$, a $2n$-sided rhombic polygon splits into $666$ rhombic quadrilaterals?

Three quadratic polynomials $f_1(x) = x^2+2a_1x+b_1$, $f_2(x) = x^2+2a_2x+b_2$, $f_3(x) = x^2 + 2a_3x + b_3$ are such that $a_1a_2a_3 = b_1b_2b_3 > 1$. Prove that at least one polynomial has two distinct roots.

Suppose $S$ is the set of all positive integers. For $a,b \in S$, define \[a * b=\frac{\text{lcm}[a,b]}{\text{gcd}(a,b)}\] For example $8*12=6$. Show that [b]exactly two[/b] of the following three properties are satisfied: (i) If $a,b \in S$, then $a*b \in S$. (ii) $(a*b)*c=a*(b*c)$ for all $a,b,c \in S$. (iii) There exists an element $i \in S$ such that $a *i =a$ for all $a \in S$.

Three pairwise intersecting rays are given. At some point in time not on every ray from its beginning a point begins to move with speed. It is known that these three points form a triangle at any time, and the center of the circumscribed circle of this the triangle also moves uniformly and on a straight line. Is it true, that all these triangles are similar to each other?

Find the smallest integer $n$ such that \[(x^2+y^2+z^2)^2\le n(x^4+y^4+z^4)\] for all real numbers $x,y,$ and $z$. $\textbf{(A) }2\qquad\textbf{(B) }3\qquad\textbf{(C) }4\qquad\textbf{(D) }6\qquad \textbf{(E) }\text{There is no such integer }n.$

1. For [i]n[/i] a positive integer, [i]n[/i]! = 1 $\cdot$ 2 $\cdot$ 3 $\dots$ ([i]n[/i] - 1) $\cdot$ [i]n[/i] is the product of the positive integers from 1 to [i]n[/i]. Determine, with proof, all positive integers [i]n[/i] for which [i]n[/i]! + 3 is a power of 3.

Find the least real number $\alpha$ such that there is a real number $\beta$ so that for all triples of real numbers $(a, b,c)$ satisfying $2006a + 10b + c = 0$, the equation $ax^2 + bx + c = 0$ always has real root in the interval $[\beta, \beta + \alpha]$.

Two circle $\Gamma_1$ and $\Gamma_2$ intersect at points $A$ and $B$, point $P$ is not on the line $AB$. Line $AP$ intersects again $\Gamma_1$ and $\Gamma_2$ at points $K$ and $L$ respectively, line $BP$ intersects again $\Gamma_1$ and $\Gamma_2$ at points $M$ and $N$ respectively and all the points mentioned so far are different. The centers of the circles circumscribed around the triangles $KMP$ and $LNP$ are $O_1$ and $O_2$ respectively. Prove that $O_1O_2$ is perpendicular to $AB$.

Let $ABC$ be an acute triangle with altitudes $AD$, $BE$, and $CF$, and let $O$ be the center of its circumcircle. Show that the segments $OA$, $OF$, $OB$, $OD$, $OC$, $OE$ dissect the triangle $ABC$ into three pairs of triangles that have equal areas.

Integers $a, b, c, d$ satisfy $a+b+c+d=0$. Show that $$n=(ab-cd)\cdot(bc-ad)\cdot(ca-bd)$$ is a perfect square.

The Stromquist Comet is visible every 61 years. If the comet is visible in 2017, what is the next leap year when the comet will be visible?

Let $ABC$ be an acute triangle with $AB = 3$ and $AC = 4$. Suppose that $AH,AO$ and $AM$ are the altitude, the bisector and the median derived from $A$, respectively. If $HO = 3 MO$, then the length of $BC$ is [img]https://cdn.artofproblemsolving.com/attachments/e/c/26cc00629f4c0ab27096b8bdc562c56ff01ce5.png[/img] A. $3$ B. $\frac72$ C. $4$ D. $\frac92$ E. $5$

In the figure, $\triangle ABC$ has $\angle A =45^{\circ}$ and $\angle B =30^{\circ}$. A line $DE$, with $D$ on $AB$ and $\angle ADE =60^{\circ}$, divides $\triangle ABC$ into two pieces of equal area. (Note: the figure may not be accurate; perhaps $E$ is on $CB$ instead of $AC$.) The ratio $\frac{AD}{AB}$ is [asy] size((220)); draw((0,0)--(20,0)--(7,6)--cycle); draw((6,6)--(10,-1)); label("A", (0,0), W); label("B", (20,0), E); label("C", (7,6), NE); label("D", (9.5,-1), W); label("E", (5.9, 6.1), SW); label("$45^{\circ}$", (2.5,.5)); label("$60^{\circ}$", (7.8,.5)); label("$30^{\circ}$", (16.5,.5)); [/asy] $ \textbf{(A)}\ \frac{1}{\sqrt{2}} \qquad\textbf{(B)}\ \frac{2}{2+\sqrt{2}} \qquad\textbf{(C)}\ \frac{1}{\sqrt{3}} \qquad\textbf{(D)}\ \frac{1}{\sqrt[3]{6}} \qquad\textbf{(E)}\ \frac{1}{\sqrt[4]{12}} $

Let $ABC$ be a triangle. The incircle of $ABC$ touches the sides $AB$ and $AC$ at the points $Z$ and $Y$, respectively. Let $G$ be the point where the lines $BY$ and $CZ$ meet, and let $R$ and $S$ be points such that the two quadrilaterals $BCYR$ and $BCSZ$ are parallelogram. Prove that $GR=GS$. [i]Proposed by Hossein Karke Abadi, Iran[/i]

Prove that there exists two different permutations $ (a_1,a_2,\dots,a_{2009})$ and $ (b_1,b_2,\dots,b_{2009})$ of $ (1,2,\dots,2009)$ such that \[ \sum_{i\equal{}1}^{2009}i^i a_i \minus{} \sum_{i\equal{}1}^{2009} i^i b_i\] is divisible by $ 2009!$.

A right triangle $A_1 A_2 A_3$ with side lengths $6,\,8,$ and $10$ on a plane $\mathcal P$ is given. Three spheres $S_1,S_2$ and $S_3$ with centers $O_1, O_2,$ and $O_3,$ respectively, are located on the same side of the plane $\mathcal P$ in such a way that $S_i$ is tangent to $\mathcal P$ at $A_i$ for $i = 1, 2, 3.$ Assume $S_1, S_2, S_3$ are pairwise externally tangent. Find the area of triangle $O_1O_2O_3.$

In a room there are $144$ people. They are joined by $n$ other people who are each carrying $k$ coins. When these coins are shared among all $n + 144$ people, each person has $2$ of these coins. Find the minimum possible value of $2n + k$.

Given a set $M$ of $1985$ positive integers, none of which has a prime divisor larger than $26$, prove that the set has four distinct elements whose geometric mean is an integer.

Let $a, b, c, d$ be reals such that $a \ge b \ge c \ge d$ and \[ (a - b)^3 + (b - c)^3 + (c - d)^3 - 2(d - a)^3 - 12(a - b)^2 - 12(b - c)^2 - 12(c - d)^2 + 12(d - a)^2 - 2020(a - b)(b - c)(c - d)(d - a) = 1536. \] Find the minimum possible value of $d - a$.

Triangle $ABC,$ with $BC = 48,$ is inscribed in a circle $\Omega$ of radius $49\sqrt{3}.$ There is a unique circle $\omega$ that is tangent to $\overline{AB}$ and $\overline{AC}$ and internally tangent to $\Omega.$ Let $D,$ $E,$ and $F$ be the points at which $\omega$ is tangent to $\Omega,$ $\overline{AB},$ and $\overline{AC},$ respectively. The rays $\overrightarrow{DE}$ and $\overrightarrow{DF}$ intersect $\Omega$ at points $X$ and $Y,$ respectively, such that $X \neq D$ and $Y \neq D.$ Compute $XY.$

Let $a,b$ be positive real numbers. Define $m(a,b)$ as the minimum of $\[ a,\frac{1}{b} \text{ and } \frac{1}{a}+b.\]$ Find the maximum of $m(a,b).$