Supppose five points in a plane are situated so that no two of the straight lines joining them are parallel, perpendicular, or coincident. From each point perpendiculars are drawn to all the lines joining the other four points. Determine the maxium number of intersections that these perpendiculars can have.

Find all integers $ x$ such that $ x(x\plus{}1)(x\plus{}7)(x\plus{}8)$ is a perfect square It's a nice problem ...hope you enjoy it! Daniel

An $n\times n$ square grid ($n\geqslant 3$) is rolled into a cylinder. Some of the cells are then colored black. Show that there exist two parallel lines (horizontal, vertical or diagonal) of cells containing the same number of black cells. [i]E. Poroshenko[/i]

To whom is the discovery of the nuclear atom attributed? $ \textbf{(A) }\text{Neils Bohr}\qquad\textbf{(B) }\text{Louis deBroglie}\qquad$ $\textbf{(C) }\text{Robert Millikan}\qquad\textbf{(D) }\text{Ernest Rutherford}\qquad $

Define the [i]hotel elevator cubic [/i]as the unique cubic polynomial $P$ for which $P(11) = 11$, $P(12) = 12$, $P(13) = 14$, $P(14) = 15$. What is $P(15)$? [i]Proposed by Evan Chen[/i]

Jeff the delivery driver starts at the point $(5, 0)$ and has to make deliveries at all the other lattice points with distance $5$ from the origin before returning to his starting point. The length of a shortest possible path he can make is $\sqrt{a}+\sqrt{b}$ for positive integers $a$ and $b.$ Find $a + b.$

Given is the sequence $(a_n)_{n\geq 0}$ which is defined as follows:$a_0=3$ and $a_{n+1}-a_n=n(a_n-1) \ , \ \forall n\geq 0$. Determine all positive integers $m$ such that $\gcd (m,a_n)=1 \ , \ \forall n\geq 0$.

Find all differentiable functions $f:\mathbb{R}\to\mathbb{R}$ such that \[f'(x)=\frac{f(x+n)-f(x)}n\] for all real numbers $x$ and all positive integers $n.$

Let $ABC$ be a triangle with $AB=AC$ and let $D$ be the midpoint of $AC$. The angle bisector of $\angle BAC$ intersects the circle through $D,B$ and $C$ at the point $E$ inside the triangle $ABC$. The line $BD$ intersects the circle through $A,E$ and $B$ in two points $B$ and $F$. The lines $AF$ and $BE$ meet at a point $I$, and the lines $CI$ and $BD$ meet at a point $K$. Show that $I$ is the incentre of triangle $KAB$. [i]Proposed by Jan Vonk, Belgium and Hojoo Lee, South Korea[/i]

Compute $1 \times 4 - 2 \times 3 + 2 \times 5 - 3 \times 4 + 3 \times 6 - 4 \times 5 + 4 \times 7 - 5 \times 6 + 5 \times 8 - 6 \times 7.$

$a>0$ $f:[-a,a]\rightarrow R$ such that $f''$ exist and Riemann-integrable suppose $f(a)=f(-a)$ $ f'(-a)=f'(a)=a^2$ Prove that $6a^3\leq \int_{-a}^{a}{f''(x)}^2dx$ Study equality case ? Radu Miculescu

You have a list of $2023$ numbers, where each one can be $-1$, $0$, $1$ or $2$. The sum of all numbers is $19$ and the sum of their squares is $99$. What are the minimum and maximum values of the sum of the cubes of those $2023$ numbers?

Determine all pairs $(x, y)$ of positive integers satisfying $x + y + 1 | 2xy$ and $ x + y - 1 | x^2 + y^2 - 1$.

Compute the remainder when $\left(\frac{2^5}{2}\right)^5$ is divided by $5$. [i]2020 CCA Math Bonanza Individual Round #3[/i]

Let $a>1$ and $c$ be natural numbers and let $b\neq 0$ be an integer. Prove that there exists a natural number $n$ such that the number $a^n+b$ has a divisor of the form $cx+1$, $x\in\mathbb{N}$.

Let $h_a, h_b, h_c$ and $r$ be the lengths of altitudes and radius of the inscribed circle of $\vartriangle ABC$, respectively. Prove that $h_a + 4h_b + 9h_c > 36r$.

Taotao wants to buy a bracelet. The bracelets have 7 different beads on them, arranged in a circle. Two bracelets are the same if one can be rotated or flipped to get the other. If she can choose the colors and placement of the beads, and the beads come in orange, white, and black, how many possible bracelets can she buy?

Let $\vartriangle ABC$ be an acute triangle with altitudes $AD$, $BE$, $CF$ and orthocenter $H$. Circle $\odot V$ is the circumcircle of $\vartriangle DE F$. Let segments $FD$, $BH$ intersect at point $P$. Let segments $ED$, $HC$ intersect at point $Q$. Let $K$ be a point on $AC$ such that $VK \perp CF$. a) Prove that $\vartriangle PQH \sim \vartriangle AKV$. b) Let line $PQ$ intersect $\odot V$ at points $I,G$. Prove that points $B,I,H,G,C$ are concyclic [hide]with center the symmetric point $X$ of circumcenter $O$ of $\vartriangle ABC$ wrt $BC$.[/hide] [hide=PS.] There is a chance that those in the hide were not wanted in the problem, as I tried to understand the wording from a solutions' video. I couldn't find the original wording pdf or picture.[/hide] [img]https://cdn.artofproblemsolving.com/attachments/c/3/0b934c5756461ff854d38f51ef4f76d55cbd95.png[/img] [url=https://www.geogebra.org/m/cjduebke]geogebra file[/url]

If $A$ and $B$ are fixed points on a given circle not collinear with centre $O$ of the circle, and if $XY$ is a variable diameter, find the locus of $P$ (the intersection of the line through $A$ and $X$ and the line through $B$ and $Y$).

There is at least one friend pair in a class of students with different names. Students in an ordered list of some of the students write the names of all their friends who are not currently written on the blackboard, in order. If each student on the list wrote at least one name on the board and the name of each student with at least one friend on the blackboard at the end of the process, call this list a $golden$ $ list$. Prove that there exists a $golden$ $ list$ such that number of students in this list is even.

$\ ABCD$ is a quadrilateral. the feet of the perpendicular from $\ D$ to $\ AB, BC$ are $\ P,Q$ respectively, and the feet of the perpendicular from $\ B$ to $\ AD,CD$ are $\ R,S$ respectively. Show that if $\angle PSR= \angle SPQ$, then $\ PR=QS$.

Consider a convex solid $ K$ in space and two parallel planes $ \epsilon _1$ and $ \epsilon _2$ on the distance $ 1$ tangent to $ K$. A plane $ \epsilon$ between $ \epsilon _1$ and $ \epsilon _2$ is on the distance $ d_1$ from $ \epsilon _1$. Find all $ d_1$ such that the part of $ K$ between $ \epsilon _1$ and $ \epsilon$ always has a volume not exceeding half the volume of $ K$.

Show that the number $n^{2} - 2^{2014}\times 2014n + 4^{2013} (2014^{2}-1)$ is not prime, where $n$ is a positive integer.

For positive integer $a \geq 2$, denote $N_a$ as the number of positive integer $k$ with the following property: the sum of squares of digits of $k$ in base a representation equals $k$. Prove that: a.) $N_a$ is odd; b.) For every positive integer $M$, there exist a positive integer $a \geq 2$ such that $N_a \geq M$.

For each subset $T$ of $S = \{1, 2, ... , 7\}$, the result $r(T)$ of T is computed as follows: the elements of $T$ are written, largest to smallest, and alternating signs $(+, -)$ starting with $+$ are put in front of each number. The value of the resulting expression is$ r(T)$. (For example, for $T =\{2, 4, 7\}$, we have $r(T) = +7 - 4 + 2 = 5$.) Compute the sum of $r(T)$ as $T$ ranges over all subsets of $S$.