Consider the obtuse-angled triangle $\triangle ABC$ and its side lengths $a,b,c$. Prove that $a^3\cos\angle A +b^3\cos\angle B + c^3\cos\angle C < abc$.

Determine all ordered pairs of real numbers $(a,b)$ such that the line $y=ax+b$ intersects the curve $y=\ln(1+x^2)$ in exactly one point.

Let $ Q$ be the centre of the inscribed circle of a triangle $ ABC.$ Prove that for any point $ P,$ \[ a(PA)^2 \plus{} b(PB)^2 \plus{} c(PC)^2 \equal{} a(QA)^2 \plus{} b(QB)^2 \plus{} c(QC)^2 \plus{} (a \plus{} b \plus{} c)(QP)^2, \] where $ a \equal{} BC, b \equal{} CA$ and $ c \equal{} AB.$

If there are integers $a,b,c$ such that $a^2+b^2+c^2-ab-bc-ca$ is divisible by a prime $p$ such that $\text{gcd}(p,\frac{a^2+b^2+c^2-ab-bc-ca}{p})=1$, then prove that there are integers $x,y,z$ such that $p=x^2+y^2+z^2-xy-yz-zx$.

Determine the least positive integer $n{}$ for which the following statement is true: the product of any $n{}$ odd consecutive positive integers is divisible by $45$.

(a) Suppose that $x_n~(n\ge0)$ is a sequence of real numbers with the property that $x_0^3+x_1^3+\ldots+x_n^3=(x_0+x_1+\ldots+x_n)^2$ for each $n\in\mathbb N$. Prove that for each $n\in\mathbb N_0$ there exists $m\in\mathbb N_0$ such that $x_0+x_1+\ldots+x_n=\frac{m(m+1)}2$. (b) For natural numbers $n$ and $p$, we define $S_{n,p}=1^p+2^p+\ldots+n^p$. Find all natural numbers $p$ such that $S_{n,p}$ is a perfect square for each $n\in\mathbb N$.

For $n > 1$, let $a_n$ be the number of zeroes that $n!$ ends with when written in base $n$. Find the maximum value of $\frac{a_n}{n}$.

Assume that the following three statements are true: $I$. All freshmen are human. $II$. All students are human. $III$. Some students think. Given the following four statements: $ \textbf{(1)}\ \text{All freshmen are students.}\qquad$ $\textbf{(2)}\ \text{Some humans think.}\qquad$ $\textbf{(3)}\ \text{No freshmen think.}\qquad$ $\textbf{(4)}\ \text{Some humans who think are not students.}$ Those which are logical consequences of I,II, and III are: $ \textbf{(A)}\ 2\qquad\textbf{(B)}\ 4\qquad\textbf{(C)}\ 2,3\qquad\textbf{(D)}\ 2,4\qquad\textbf{(E)}\ 1,2 $

Let $m$ and $n$ be positive integers. A circular necklace contains $mn$ beads, each either red or blue. It turned out that no matter how the necklace was cut into $m$ blocks of $n$ consecutive beads, each block had a distinct number of red beads. Determine, with proof, all possible values of the ordered pair $(m, n)$. [i]Proposed by Rishabh Das[/i]

Let $ABCD$ be a trapezoid such that $AB||CD$ and let $P=AC\cap BD,AB=21,CD=7,AD=13,[ABCD]=168.$ Let the line parallel to $AB$ through $P$ intersect the circumcircle of $BCP$ in $X.$ Circumcircles of $BCP$ and $APD$ intersect at $P,Y.$ Let $XY\cap BC=Z.$ If $\angle ADC$ is obtuse, then $BZ=\frac{a}{b},$ where $a,b$ are coprime positive integers. Compute $a+b.$

We have some identical paper squares which are black on one side of the sheet and white on the other side. We can join nine squares together to make a $3$ by $3$ sheet of squares by placing each of the nine squares either white side up or black side up. Two of these $3$ by $3$ sheets are distinguishable if neither can be made to look like the other by rotating the sheet or by turning it over. How many distinguishable $3$ by $3$ squares can we form?

The sequence $\{a_n\}$ is defined as follows: \begin{align*} a_n = \sqrt{1 + \left(1 + \frac 1n \right)^2} + \sqrt{1 + \left(1 - \frac 1n \right)^2}. \end{align*} What is the value of the expression given below? \begin{align*} \frac{4}{a_1} + \frac{4}{a_2} + \dots + \frac{4}{a_{96}}.\end{align*} $\textbf{(A)}\ \sqrt{18241} \qquad\textbf{(B)}\ \sqrt{18625} - 1 \qquad\textbf{(C)}\ \sqrt{18625} \qquad\textbf{(D)}\ \sqrt{19013} - 1\qquad\textbf{(E)}\ \sqrt{19013}$

In the grid to the right, the shortest path through unit squares between between the pair of 2's has length 2. Fill in some of the unit squares in the grid so that (i) exactly half of the squares in each row and column contain a number, (ii) each of the number 1 through 12 appears exactly twice, and (iii) for $n=1,2,\cdot\cdot\cdot,12$, the shortest path between the pair of $n$'s has length exactly $n$.

A circle with center $ O$ is tangent to the coordinate axes and to the hypotenuse of the $ 30^\circ$-$ 60^\circ$-$ 90^\circ$ triangle $ ABC$ as shown, where $ AB \equal{} 1$. To the nearest hundredth, what is the radius of the circle? [asy]defaultpen(linewidth(.8pt)); dotfactor=3; pair A = origin; pair B = (1,0); pair C = (0,sqrt(3)); pair O = (2.33,2.33); dot(A);dot(B);dot(C);dot(O); label("$A$",A,SW);label("$B$",B,SE);label("$C$",C,W);label("$O$",O,NW); label("$1$",midpoint(A--B),S);label("$60^\circ$",B,2W + N); draw((3,0)--A--(0,3)); draw(B--C); draw(Arc(O,2.33,163,288.5));[/asy]$ \textbf{(A)}\ 2.18\qquad \textbf{(B)}\ 2.24\qquad \textbf{(C)}\ 2.31\qquad \textbf{(D)}\ 2.37\qquad \textbf{(E)}\ 2.41$

[i]Centered hexagonal numbers[/i] are the numbers of dots used to create hexagonal arrays of dots. The first four centered hexagonal numbers are 1, 7, 19, and 37 as shown below: [asy] size(250);defaultpen(linewidth(0.4)); dot(origin^^shift(-12,0)*origin^^shift(-24,0)*origin^^shift(-36,0)*origin); int i; for(i=0; i<360; i=i+60) { dot(1*dir(i)^^2*dir(i)^^3*dir(i)); dot(shift(1/2, sqrt(3)/2)*1*dir(i)^^shift(1/2, sqrt(3)/2)*2*dir(i)); dot(shift(1, sqrt(3))*1*dir(i)); dot(shift(-12,0)*origin+1*dir(i)^^shift(-12,0)*origin+2*dir(i)); dot(shift(-12,0)*origin+sqrt(3)*dir(i+30)); dot(shift(-24,0)*origin+1*dir(i)); } label("$1$", (-36, -5), S); label("$7$", (-24, -5), S); label("$19$", (-12, -5), S); label("$37$", (0, -5), S); label("Centered Hexagonal Numbers", (-18,-10), S);[/asy] Consider an arithmetic sequence 1, $a$, $b$ and a geometric sequence 1,$c$,$d$, where $a$,$b$,$c$, and $d$ are all positive integers and $a+b=c+d$. Prove that each centered hexagonal number is a possible value of $a$, and prove that each possible value of $a$ is a centered hexagonal number.

Melanie computes the mean $\mu$, the median $M$, and the modes of the $365$ values that are the dates in the months of $2019$. Thus her data consist of $12$ $1\text{s}$, $12$ $2\text{s}$, . . . , $12$ $28\text{s}$, $11$ $29\text{s}$, $11$ $30\text{s}$, and $7$ $31\text{s}$. Let $d$ be the median of the modes. Which of the following statements is true? $\textbf{(A) } \mu < d < M \qquad\textbf{(B) } M < d < \mu \qquad\textbf{(C) } d = M =\mu \qquad\textbf{(D) } d < M < \mu \qquad\textbf{(E) } d < \mu < M$

Consider $64$ distinct natural numbers, at most equal to $2012$. Show that it is possible to choose four of them, denoted as $a,b,c,d$ such that $ a+b-c-d$ to be a multiple of $2013$