Let $ u_1, u_2, \ldots, u_m$ be $ m$ vectors in the plane, each of length $ \leq 1,$ with zero sum. Show that one can arrange $ u_1, u_2, \ldots, u_m$ as a sequence $ v_1, v_2, \ldots, v_m$ such that each partial sum $ v_1, v_1 \plus{} v_2, v_1 \plus{} v_2 \plus{} v_3, \ldots, v_1, v_2, \ldots, v_m$ has length less than or equal to $ \sqrt {5}.$

Consider the square $ABCD$ and the point $E$ inside the angle $CAB$, such that $\angle BAE =15^o$, and the lines $BE$ and $BD$ are perpendicular. Prove that $AE = BD$.

$\textbf{Bake Sale}$ Four friends, Art, Roger, Paul and Trisha, bake cookies, and all cookies have the same thickness. The shapes of the cookies differ, as shown. $\circ$ Art's cookies are trapezoids: [asy]size(80);defaultpen(linewidth(0.8));defaultpen(fontsize(8)); draw(origin--(5,0)--(5,3)--(2,3)--cycle); draw(rightanglemark((5,3), (5,0), origin)); label("5 in", (2.5,0), S); label("3 in", (5,1.5), E); label("3 in", (3.5,3), N);[/asy] $\circ$ Roger's cookies are rectangles: [asy]size(80);defaultpen(linewidth(0.8));defaultpen(fontsize(8)); draw(origin--(4,0)--(4,2)--(0,2)--cycle); draw(rightanglemark((4,2), (4,0), origin)); draw(rightanglemark((0,2), origin, (4,0))); label("4 in", (2,0), S); label("2 in", (4,1), E);[/asy] $\circ$ Paul's cookies are parallelograms: [asy]size(80);defaultpen(linewidth(0.8));defaultpen(fontsize(8)); draw(origin--(3,0)--(2.5,2)--(-0.5,2)--cycle); draw((2.5,2)--(2.5,0), dashed); draw(rightanglemark((2.5,2),(2.5,0), origin)); label("3 in", (1.5,0), S); label("2 in", (2.5,1), W);[/asy] $\circ$ Trisha's cookies are triangles: [asy]size(80);defaultpen(linewidth(0.8));defaultpen(fontsize(8)); draw(origin--(3,0)--(3,4)--cycle); draw(rightanglemark((3,4),(3,0), origin)); label("3 in", (1.5,0), S); label("4 in", (3,2), E);[/asy] Each friend uses the same amount of dough, and Art makes exactly 12 cookies. Art's cookies sell for 60 cents each. To earn the same amount from a single batch, how much should one of Roger's cookies cost in cents? $ \textbf{(A)}\ 18\qquad\textbf{(B)}\ 25\qquad\textbf{(C)}\ 40\qquad\textbf{(D)}\ 75\qquad\textbf{(E)}\ 90$

Let $O$ be the center and $R$ be the radius of circumcircle $\omega$ of triangle $ABC$. Circle $\omega_1$ with center $O_1$ and radius $R$ pass through points $A, O$ and intersects the side $AC$ at point $K$. Let $AF$ be the diameter of circle $\omega$ and points $F, K, O_1$ are collinear. Determine $\angle ABC$:

(a) $K_1,K_2,..., K_n$ are the feet of the perpendiculars from an arbitrary point $M$ inside a given regular $n$-gon to its sides (or sides produced). Prove that the sum $\overrightarrow{MK_1} + \overrightarrow{MK_2} + ... + \overrightarrow{MK_n}$ equals $\frac{n}{2}\overrightarrow{MO}$, where $O$ is the centre of the $n$-gon. (b) Prove that the sum of the vectors whose origin is an arbitrary point $M$ inside a given regular tetrahedron and whose endpoints are the feet of the perpendiculars from $M$ to the faces of the tetrahedron equals $\frac43 \overrightarrow{MO}$, where $O$ is the centre of the tetrahedron. (VV Prasolov, Moscow)

Let $n\geq 2$ be an integer and $a_1,a_2,\cdots,a_n>1$ be real numbers. Prove that the inequality below holds. $$\prod_{i=1}^n\left(a_ia_{i+1}-\frac{1}{a_ia_{i+1}}\right)\geq 2^n\prod_{i=1}^n\left(a_i-\frac{1}{a_i}\right)$$

Suppose that $a_1, a_2, a_3, \ldots$ is an infinite geometric sequence such that for all $i \ge 1$, $a_i$ is a positive integer. Suppose furthermore that $a_{20} + a_{21} = 20^{21}$. If the minimum possible value of $a_1$ can be expressed as $2^a 5^b$ for positive integers $a$ and $b$, find $a + b$. [i]Proposed by Andrew Wu[/i]

Let $ ABC$ be a triangle with $ AB \equal{} AC$. A ray $ Ax$ is constructed in space such that the three planar angles of the trihedral angle $ ABCx$ at its vertex $ A$ are equal. If a point $ S$ moves on $ Ax$, find the locus of the incenter of triangle $ SBC$.

Find the value of $x+y$ for which the expression \[\frac{6x^2}{y^6} + \frac{6y^2}{x^6}+9x^2y^2+\frac{4}{x^6y^6}\] is minimized.

Is there a closed self-intersecting broken line in space that intersects each of its links exactly once, and in its midpoint ?

In $\triangle ABC$, $P,Q,R$ divides the perimeter of $\triangle ABC$ into three equal parts. $P,Q\in AB$. Prove that $\frac{S_{\triangle PQR}}{S_{\triangle ABC}}>\frac{2}{9}$.

Given $\frac{\log{a}}{p}=\frac{\log{b}}{q}=\frac{\log{c}}{r}=\log{x}$, all logarithms to the same base and $x \not= 1$. If $\frac{b^2}{ac}=x^y$, then $y$ is: $ \text{(A)}\ \frac{q^2}{p+r}\qquad\text{(B)}\ \frac{p+r}{2q}\qquad\text{(C)}\ 2q-p-r\qquad\text{(D)}\ 2q-pr\qquad\text{(E)}\ q^2-pr$

The numbers $3$, $4$ ,$...$, $2019$ are written on a blackboard. David and Edgardo play alternately, starting with David. On their turn, each player must erase a number from the board and write two positive integers whose sum is equal to the number just deleted. The winner is the one who makes all the numbers on the board equal. Determine who has a winning strategy and describe it.

Suppose a $3\times 3$ matrix $A$ satisfies $\mathbf{v}^t A \mathbf{v} > 0$ for any vector $\mathbf{v} \in\mathbb{R}^3 -\{0\}$. (Note that $A$ may not be a symmetric matrix.) (1) Prove that $\det(A)>0$. (2) Consider diagonal matrix $D=\text{diag}(-1,1,1)$. Prove that there's exactly one negative real among eigenvalues of $AD$.

Find the minimum area bounded by the graphs of $ y\equal{}x^2$ and $ y\equal{}kx(x^2\minus{}k)\ (k>0)$.

If $a$ and $b$ are integers and if the solutions of the system of equations $$y - 2x - a = 0$$ $$y^2 - xy + x^2 - b = 0$$ are rational, prove that the solutions are integers.

Consider a cyclic quadrilateral $ABCD$. The extensions of its sides $AB,DC$ meet at the point $P$ and the extensions of its sides $AD,BC$ meet at the point $Q$. Suppose $\quad QE,QF$ are tangents to the circumcircle of quadrilateral $ABCD$ at $E,F$ respectively. Show that $P,E,F$ are collinear.

Let $x_1,x_2,\ldots,x_n$ be arbitrary real numbers. Prove the inequality \[ \frac{x_1}{1+x_1^2} + \frac{x_2}{1+x_1^2 + x_2^2} + \cdots + \frac{x_n}{1 + x_1^2 + \cdots + x_n^2} < \sqrt{n}. \]

Let \(f:\mathbb R\to\mathbb R\) be a function such that for all real numbers \(x\neq1\), \[f(x-f(x))+f(x)=\frac{x^2-x+1}{x-1}.\] Find all possible values of \(f(2023)\). [i]Proposed by Linus Tang[/i]

Let $ D$, $ E$, $ F$ be the feet of the altitudes wrt sides $ BC$, $ CA$, $ AB$ of acute-angled triangle $ \triangle ABC$, respectively. In triangle $ \triangle CFB$, let $ P$ be the foot of the altitude wrt side $ BC$. Define $ Q$ and $ R$ wrt triangles $ \triangle ADC$ and $ \triangle BEA$ analogously. Prove that lines $ AP$, $ BQ$, $ CR$ don't intersect in one common point.

How many integers $0 \leq x < 125$ are there such that $x^3 - 2x + 6 \equiv  0 \pmod {125}$? $ \textbf{a)}\ 0 \qquad\textbf{b)}\ 1 \qquad\textbf{c)}\ 2 \qquad\textbf{d)}\ 3 \qquad\textbf{e)}\ \text{None of above} $

On an $m\times n$ grid there's a snail in each cell. Each round, the snail army can choose four snail in a $2\times 2$ square and make them perform the complete [b]Quadrilateral Dance[/b], which is rotating the four snails clockwise by $90$ degrees, moving each one to an adjacent cell. Find all pairs of positive integers $(m,n)$ such that the snails can achieve any permutation by performing a finite number of times of [b]Quadrilateral Dance[/b]. [i]Proposed by chengbilly[/i]

In the tetrahedron $ABCD,\angle BDC=90^o$ and the foot of the perpendicular from $D$ to $ABC$ is the intersection of the altitudes of $ABC$. Prove that: \[ (AB+BC+CA)^2\le6(AD^2+BD^2+CD^2). \] When do we have equality?

Find the biggest positive integer $n$ for which the number $(n!)^6-6^n$ is divisible by $2022$.

A 16 ×16 square sheet of paper is folded once in half horizontally and once in half vertically to make an 8 × 8 square. This square is again folded in half twice to make a 4 × 4 square. This square is folded in half twice to make a 2 × 2 square. This square is folded in half twice to make a 1 × 1 square. Finally, a scissor is used to make cuts through both diagonals of all the layers of the 1 × 1 square. How many pieces of paper result?