A circle and a point $P$ higher than the circle lie in the same vertical plane. A particle moves along a straight line under gravity from $P$ to a point $Q$ on the circle. Given that the distance travelled from $P$ in time $t$ is equal to $\dfrac{1}{2}gt^2 \sin{\alpha}$, where $\alpha$ is the angle of inclination of the line $PQ$ to the horizontal, give a geometrical characterization of the point $Q$ for which the time taken from $P$ to $Q$ is a minimum.

Let $ H$ be the orthocenter, $ O$ the circumcenter, and $ R$ the circumradius of an acute-angled triangle $ ABC$. Consider the circles $ k_a,k_b,k_c,k_h,k$, all with radius $ R$, centered at $ A,B,C,H,M,$ respectively. Circles $ k_a$ and $ k_b$ meet at $ M$ and $ F$; $ k_a$ and $ k_c$ meet at $ M$ and $ E$; and $ k_b$ and $ k_c$ meet at $ M$ and $ D$. $ (a)$ Prove that the points $ D,E,F$ lie on the circle $ k_h$. $ (b)$ Prove that the set of the points inside $ k_h$ that are inside exactly one of the circles $ k_a,k_b,k_c$ has the area twice the area of $ \triangle ABC$.

Prove that for any three infinite sequences of natural numbers $(a_n)_{n\ge 1}$, $(b_n)_{n\ge 1}$, $(c_n)_{n\ge 1}$, there exist numbers $p$ and $q$ such that $a_p\ge a_q$, $b_p\ge b_q$ and $c_p\ge c_q$.

Let $p$ be a prime number greater than $2$. Patricia wants $7$ not-necessarily different numbers from $\{1, 2, . . . , p\}$ to the black dots in the figure below, on such a way that the product of three numbers on a line or circle always has the same remainder when divided by $p$. [img]https://cdn.artofproblemsolving.com/attachments/3/1/ef0d63b8ff5341ffc340de0cc75b24c7229e23.png[/img] (a) Suppose Patricia uses the number $p$ at least once. How many times does she have the number $p$ then a minimum sum needed? (b) Suppose Patricia does not use the number $p$. In how many ways can she assign numbers? (Two ways are different if there is at least one black one dot different numbers are assigned. The figure is not rotated or mirrored.)

A rectangle $ABCD$ is inscribed in a circle with centre $O$. The exterior bisectors of $\angle ABD$ and $\angle ADB$ intersect at $P$; those of $\angle DAB$ and $\angle DBA$ intersect at $Q$; those of $\angle ACD$ and $\angle ADC$ intersect at $R$; and those of $\angle DAC$ and $\angle DCA$ intersect at $S$. Prove that $P,Q,R$, and $S$ are concyclic.

A solid rectangular block is formed by gluing together $N$ congruent 1-cm cubes face to face. When the block is viewed so that three of its faces are visible, exactly 231 of the 1-cm cubes cannot be seen. Find the smallest possible value of $N$.

Let $f_n(x)=\sin^n x + \cos^n x$. For how many $x$ in $[0,\pi]$ is it true that \[6f_4(x)-4f_6(x)=2f_2(x)?\] $\textbf{(A) }2\qquad\textbf{(B) }4\qquad\textbf{(C) }6\qquad\textbf{(D) }8\qquad\textbf{(E) }\text{more than 8}$

Elmo bakes cookies at a rate of one per 5 minutes. Big Bird bakes cookies at a rate of one per 6 minutes. Cookie Monster [i]consumes[/i] cookies at a rate of one per 4 minutes. Together Elmo, Big Bird, Cookie Monster, and Oscar the Grouch produce cookies at a net rate of one per 8 minutes. How many minutes does it take Oscar the Grouch to bake one cookie?

Rational numbers $a, b$ are such that $a+b$ and $a^2+b^2$ are integers. Prove that $a, b$ are integers.

Find all positive integers $a$ such that for any positive integer $n\ge 5$ we have $2^n-n^2\mid a^n-n^a$.

If in applying the quadratic formula to a quadratic equation \[ f(x)\equiv ax^2 \plus{} bx \plus{} c \equal{} 0, \] it happens that $ c \equal{} \frac {b^2}{4a}$, then the graph of $ y \equal{} f(x)$ will certainly: $ \textbf{(A)}\ \text{have a maximum} \qquad\textbf{(B)}\ \text{have a minimum} \qquad\textbf{(C)}\ \text{be tangent to the x \minus{} axis} \\ \qquad\textbf{(D)}\ \text{be tangent to the y \minus{} axis} \qquad\textbf{(E)}\ \text{lie in one quadrant only}$

Let ${f}$ be a bounded real function defined for all real numbers and satisfying for all real numbers ${x}$ the condition ${ f \Big(x+\frac{1}{3}\Big) + f \Big(x+\frac{1}{2}\Big)=f(x)+ f \Big(x+\frac{5}{6}\Big)}$ . Show that ${f}$ is periodic.

Let $n$ be the square of an integer whose each prime divisor has an even number of decimal digits. Consider $P(x) = x^n - 1987x$. Show that if $x,y$ are rational numbers with $P(x) = P(y)$, then $x = y$.

Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that $f(x)f(y)=f(x+y)+xy$ for all $x,y\in \mathbb{R}$.

Let $f:\mathbb{R}\to\mathbb{R}$ be a bijective function. Does there always exist an infinite number of functions $g:\mathbb{R}\to\mathbb{R}$ such that $f(g(x))=g(f(x))$ for all $x\in\mathbb{R}$? [i]Proposed by Daniel Liu[/i]

Show that among any $n\geq 3$ vertices of a regular $(2n-1)$-gon we can find $3$ of them forming an isosceles triangle.

Find all pairs of positive integers $m$, $n$ such that the $(m+n)$-digit number \[\underbrace{33\ldots3}_{m}\underbrace{66\ldots 6}_{n}\] is a perfect square.

A fly trapped inside a cubical box with side length $ 1$ meter decides to relieve its boredom by visiting each corner of the box. It will begin and end in the same corner and visit each of the other corners exactly once. To get from a corner to any other corner, it will either fly or crawl in a straight line. What is the maximum possible length, in meters, of its path? $ \textbf{(A)}\ 4 \plus{} 4\sqrt2 \qquad \textbf{(B)}\ 2 \plus{} 4\sqrt2 \plus{} 2\sqrt3 \qquad \textbf{(C)}\ 2 \plus{} 3\sqrt2 \plus{} 3\sqrt3 \qquad \textbf{(D)}\ 4\sqrt2 \plus{} 4\sqrt3 \\ \textbf{(E)}\ 3\sqrt2 \plus{} 5\sqrt3$

A bag of popping corn contains $\frac{2}{3}$ white kernels and $\frac{1}{3}$ yellow kernels. Only $\frac{1}{2}$ of the white kernels will pop, whereas $\frac{2}{3}$ of the yellow ones will pop. A kernel is selected at random from the bag, and pops when placed in the popper. What is the probability that the kernel selected was white? $ \textbf{(A)}\ \frac{1}{2} \qquad\textbf{(B)}\ \frac{5}{9} \qquad\textbf{(C)}\ \frac{4}{7} \qquad\textbf{(D)}\ \frac{3}{5} \qquad\textbf{(E)}\ \frac{2}{3} $

Find all functions $f : \mathbb{R} \to \mathbb{R}$ such that for all real numbers $a, b, c$, we have \[ f(a+b+c)f(ab+bc+ca) - f(a)f(b)f(c) = f(a+b)f(b+c)f(c+a). \] [i]Proposed by Mainak Ghosh and Rijul Saini[/i]

We are allowed to put several brackets in the expression $$\frac{29 : 28 : 27 : 26 :... : 17 : 16}{15 : 14 : 13 : 12 : ... : 3 : 2}$$ always in the same places below each other. (a) Find the smallest possible integer value we can obtain in that way. (b) Find all possible integer values that can be obtained. Remark: in this problem, $$\frac{(29 : 28) : 27 : ... : 16}{(15 : 14) : 13 : ... : 2},$$ is valid position of parenthesis, on the other hand $$\frac{(29 : 28) : 27 : ... : 16}{15 : (14 : 13) : ... : 2}$$ is forbidden.

Each of the natural numbers from $1$ to $n$ is colored either red or blue, with each color being used at least once. It turns out that: – every red number is a sum of two distinct blue numbers; and – every blue number is a difference between two red numbers. Determine the smallest possible value of $n$ for which such a coloring exists.

Let $n$ be a positive integer. Define a chameleon to be any sequence of $3n$ letters, with exactly $n$ occurrences of each of the letters $a, b,$ and $c$. Define a swap to be the transposition of two adjacent letters in a chameleon. Prove that for any chameleon $X$ , there exists a chameleon $Y$ such that $X$ cannot be changed to $Y$ using fewer than $3n^2/2$ swaps.

Prove that the numbers $A,B$ and $C$ are equal, where: - $A=$ number of ways that we can cover a $2 \times n$ rectangle with $2 \times 1$ retangles. - $B=$ number of sequences of ones and twos that add up to $n$ - $C= \sum^m_{k=0} \binom{m + k}{2 \cdot k}$ if $n = 2 \cdot m,$ and - $C= \sum^m_{k=0} \binom{m + k + 1}{2 \cdot k + 1}$ if $n = 2 \cdot m + 1.$

Let's consider a set of distinct positive integers with a sum equal to 2023. Among these integers, there are a total of $d$ even numbers and $m$ odd numbers. Determine the maximum possible value of $2d + 4m$. [i]Proposed by Gogi Khimshiashvili, Georgia[/i]