Consider pairs of functions $(f, g)$ from the set of nonnegative integers to itself such that [list] [*] $f(0) + f(1) + f(2) + \cdots + f(42) \le 2022$; [*] for any integers $a \ge b \ge 0$, we have $g(a+b) \le f(a) + f(b)$. [/list] Determine the maximum possible value of $g(0) + g(1) + g(2) + \cdots + g(84)$ over all such pairs of functions. [i]Evan Chen (adapting from TST3, by Sean Li)[/i]

$N$ students are seated at desks in an $m \times n$ array, where $m, n \ge 3$. Each student shakes hands with the students who are adjacent horizontally, vertically or diagonally. If there are $1020 $handshakes, what is $N$?

If four times the reciprocal of the circumference of a circle equals the diameter of the circle, then the area of the circle is $\textbf{(A) }\frac{1}{\pi^2}\qquad\textbf{(B) }\frac{1}{\pi}\qquad\textbf{(C) }1\qquad\textbf{(D) }\pi\qquad \textbf{(E) }\pi^2$

A geometry program on the computer allows the following operations to be performed: [list] [*]Mark points on segments, on lines or outside them. [*]Draw the line that joins two points. [*]Find the point of intersection of two lines. [*]Given a point $P$ and a line $\ell$, trace the symmetric of $P$ with respect to $\ell$. [/list] Given an triangle $ABC$, using exclusively the allowed operations, construct the intersection point of the perpendicular bisectors of the triangle.

Let $\triangle ABC$ be a triangle with $AB=10$ and $AC=16,$ and let $I$ be the intersection of the internal angle bisectors of $\triangle ABC.$ Suppose the tangents to the circumcircle of $\triangle BIC$ at $B$ and $C$ intersect at a point $P$ with $PA=8.$ Compute the length of ${BC}.$ [i]Proposed by Kyle Lee[/i]

Let $p$ be a prime number of the form $4k+1$. Suppose that $2p+1$ is prime. Show that there is no $k \in \mathbb{N}$ with $k<2p$ and $2^k \equiv 1 \; \pmod{2p+1}$.

Given a triangle $ABC$ with $AB = AC$, angle $\angle A = 100^o$ and $BD$ bisector of angle $\angle B$. Prove that $$BC = BD + DA.$$

Let $z$ be a complex number such that $|z| = 1$ and $|z-1.45|=1.05$. Compute the real part of $z$.

The base of the quadrangular pyramid $SABCD$ is a quadrilateral $ABCD$, the diagonals of which are perpendicular. The apex of the pyramid is projected at intersection point $O$ of the diagonals of the base. Prove that the feet of the perpendiculars drawn from point $O$ to the side faces of the pyramid lie on one circle.

Let $ABCD$ be a cyclic quadrilateral satisfying $\angle ABD = \angle DBC$. Let $E$ be the intersection of the diagonals $AC$ and $BD$. Let $M$ be the midpoint of $AE$, and $N$ be the midpoint of $DC$. Show that $MBCN$ is a cyclic quadrilateral.

Given distinct prime numbers $p_1,\ldots,p_s$ and a positive integer $n$, find the number of positive integers not exceeding $n$ that are divisible by exactly one of the $p_i$.

In a triangle $ABC$ we have $|AB| = |AC|$ and $\angle BAC = \alpha$. Let $P \ne B$ be a point on $AB$ and $Q$ a point on the altitude drawn from $A$ such that $|PQ| = |QC|$. Find $ \angle QPC$.

Which of the following is a possible number of diagonals of a convex polygon? (A) $02$ (B) $21$ (C) $32$ (D) $54$ (E) $63$

$ABCD$ is a cyclic quadrilateral with circumcenter $O$ and circumradius $7$. $AB$ intersects $CD$ at $E$, $DA$ intersects $CB$ at $F$. $OE=13$, $OF=14$. Let $\cos\angle FOE=\dfrac pq$, with $p$, $q$ coprime. Find $p+q$.

A trapezoid $ABCD$ with bases $BC$ and $AD$ is given. The points $K$ and $L$ are chosen on the sides $AB$ and $CD$, respectively, so that $KL \parallel AD$. It turned out that the areas of the quadrilaterals $AKLD$ and $KBCL$ are equal. Find the length $KL$ if $BC = 3, AD = 5$.

Let $\mathbb{Z}^n$ be the integer lattice in $\mathbb{R}^n$. Two points in $\mathbb{Z}^n$ are called {\em neighbors} if they differ by exactly 1 in one coordinate and are equal in all other coordinates. For which integers $n \geq 1$ does there exist a set of points $S \subset \mathbb{Z}^n$ satisfying the following two conditions? \\ (1) If $p$ is in $S$, then none of the neighbors of $p$ is in $S$. \\ (2) If $p \in \mathbb{Z}^n$ is not in $S$, then exactly one of the neighbors of $p$ is in $S$.

An "n-pointed star" is formed as follows: the sides of a convex polygon are numbered consecutively $1,2,\cdots,k,\cdots,n$, $n\geq 5$; for all $n$ values of $k$, sides $k$ and $k+2$ are non-parallel, sides $n+1$ and $n+2$ being respectively identical with sides $1$ and $2$; prolong the $n$ pairs of sides numbered $k$ and $k+2$ until they meet. (A figure is shown for the case $n=5$) [img]http://www.artofproblemsolving.com/Forum/album_pic.php?pic_id=704&sid=8da93909c5939e037aa99c429b2d157a[/img] Let $S$ be the degree-sum of the interior angles at the $n$ points of the star; then $S$ equals: $\text{(A)} \ 180 \qquad \text{(B)} \ 360 \qquad \text{(C)} \ 180(n+2) \qquad \text{(D)} \ 180(n-2) \qquad \text{(E)} \ 180(n-4)$

In an acute $\triangle ABC$ ($AB \neq AC$) with angle $\alpha$ at the vertex $A$, point $E$ is the nine-point center, and $P$ a point on the segment $AE$. If $\angle ABP = \angle ACP = x$, prove that $x = 90$° $ -2 \alpha $. [i]Proposed by Dusan Djukic[/i]

Which one below cannot be expressed in the form $x^2+y^5$, where $x$ and $y$ are integers? $ \textbf{(A)}\ 59170 \qquad\textbf{(B)}\ 59149 \qquad\textbf{(C)}\ 59130 \qquad\textbf{(D)}\ 59121 \qquad\textbf{(E)}\ 59012 $

There are 300 apples, any two of which differ in weight by no more than three times. Prove that they can be arranged into bags of four apples each so that any two bags differ in weight by no more than than one and a half times.

A rectangle is drawn on checkered paper, the sides of which form angles of $45^o$ with the grid lines, and the vertices do not lie on the grid lines. Can an odd number of grid lines intersect each side of a rectangle?

Let $a,b,c$ be sides of a triangle. Show that $a^3+b^3+3abc>c^3$.

Let $A=\{1,2,3,\ldots,40\}$. Find the least positive integer $k$ for which it is possible to partition $A$ into $k$ disjoint subsets with the property that if $a,b,c$ (not necessarily distinct) are in the same subset, then $a\ne b+c$.

On a plane, two equilateral triangles (of side length $1$) share a side, and a circle is drawn with the common side as a diameter. Find the area of the set of all points that lie inside exactly one of these shapes. [i]Proposed by Howard Halim[/i]

Find all prime numbers whose sixth power does not give remainder $1$ when dividing by $504$