Given two vectors $\overrightarrow a$, $\overrightarrow b$, find the range of possible values of $\|\overrightarrow a - 2 \overrightarrow b\|$ where $\|\overrightarrow v\|$ denotes the magnitude of a vector $\overrightarrow v$. [i](Source: China National High School Mathematics League 2021, Zhejiang Province, Problem 1)[/i]

XOY is angle in the plane.A,B are variable point on OX,OY such that 1/OA+1/OB=1/K (k is constant).draw two circles with diameter OA and OB.prove that common external tangent to these circles is tangent to the constant circle( ditermine the radius and the locus of its center).

Let $ n$ be an even number, and $ S$ be the set of all arrays of length $ n$ whose elements are from the set $ \left\{0,1\right\}$. Prove that $ S$ can be partitioned into disjoint three-element subsets such that for each three arrays $ \left(a_i\right)_{i \equal{} 1}^n$, $ \left(b_i\right)_{i \equal{} 1}^n$, $ \left(c_i\right)_{i \equal{} 1}^n$ which belong to the same subset and for each $ i\in\left\{1,2,...,n\right\}$, the number $ a_i \plus{} b_i \plus{} c_i$ is divisible by $ 2$.

Let $ABCD$ be a quadrilateral and $M$ the midpoint of the segment $AB$. Outside of the quadrilateral are constructed the equilateral triangles $BCE$, $CDF$ and $DAG$. Let $P$ and $N$ be the midpoints of the segments $GF$ and $EF$. Prove that the triangle $MNP$ is equilateral.

Find the minimum possible length of the sum of $1999$ unit vectors in the coordinate plane whose both coordinates are nonnegative.

Let $H$ be the orthocenter of the acute triangle $ABC.$ In the plane of the triangle $ABC$ we consider a point $X$ such that the triangle $XAH$ is right and isosceles, having the hypotenuse $AH,$ and $B$ and $X$ are on each part of the line $AH.$ Prove that $\overrightarrow{XA}+\overrightarrow{XC}+\overrightarrow{XH}=\overrightarrow{XB}$ if and only if $ \angle BAC=45^{\circ}.$

Let $ABC$ be an acute triangle, and let $M$ and $N$ be two points on the line $AC$ such that the vectors $MN$ and $AC$ are identical. Let $X$ be the orthogonal projection of $M$ on $BC$, and let $Y$ be the orthogonal projection of $N$ on $AB$. Finally, let $H$ be the orthocenter of triangle $ABC$. Show that the points $B$, $X$, $H$, $Y$ lie on one circle.

Three equilateral triangles $ABC, CDE, EHK$ (the vertices are mentioned counterclockwise) are lying in the plane so, that the vectors $\overrightarrow{AD}$ and $\overrightarrow{DK}$ are equal. Prove that the triangle $BHD$ is also equilateral

Let $ABCD$ and $A^{\prime }B^{\prime}C^{\prime }D^{\prime }$ be two arbitrary parallelograms in the space, and let $M,$ $N,$ $P,$ $Q$ be points dividing the segments $AA^{\prime },$ $BB^{\prime },$ $CC^{\prime },$ $DD^{\prime }$ in equal ratios. [b]a.)[/b] Prove that the quadrilateral $MNPQ$ is a parallelogram. [b]b.)[/b] What is the locus of the center of the parallelogram $MNPQ,$ when the point $M$ moves on the segment $AA^{\prime }$ ? (Consecutive vertices of the parallelograms are labelled in alphabetical order.

Let $\xi (E, \pi, B)\, (\pi\colon E\rightarrow B)$ be a real vector bundle of finite rank, and let $$\tau_E=V\xi \oplus H\xi\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, (*)$$ be the tangent bundle of $E$, where $V\xi=\mathrm{Ker}\, d\pi$ is the vertical subbundle of $\tau_E$. Let us denote the projection operators corresponding to the splitting $(*)$ by $v$ and $h$. Construct a linear connection $\nabla$ on $V\xi$ such that $$\nabla_X\lor Y - \nabla_Y \lor X=v[X,Y] - v[hX,hY]$$ ($X$ and $Y$ are vector fields on $E$, $[.,\, .]$ is the Lie bracket, and all data are of class $\mathcal C^\infty$. [J. Szilasi]

Let $ABCD$ be a convex quadrilateral such that $AC \perp BD$. (a) Prove that $AB^2 + CD^2 = BC^2 + DA^2$. (b) Let $PQRS$ be a convex quadrilateral such that $PQ = AB$, $QR = BC$, $RS = CD$ and $SP = DA$. Prove that $PR \perp QS$.

Suppose $a_1, \dots, a_n$ are integers whose greatest common divisor is 1. Let $S$ be a set of integers with the following properties: (a) For $i=1, \dots, n$, $a_i \in S$. (b) For $i,j = 1, \dots, n$ (not necessarily distinct), $a_i - a_j \in S$. (c) For any integers $x,y \in S$, if $x+y \in S$, then $x-y \in S$. Prove that $S$ must be equal to the set of all integers.

$H,I,O,N$ are orthogonal center, incenter, circumcenter, and Nagelian point of triangle $ABC$. $I_{a},I_{b},I_{c}$ are excenters of $ABC$ corresponding vertices $A,B,C$. $S$ is point that $O$ is midpoint of $HS$. Prove that centroid of triangles $I_{a}I_{b}I_{c}$ and $SIN$ concide.

Let $ a,b,c$ be positive real numbers. Prove the inequality: $ \frac {a^3}{b^2} \plus{} \frac {b^3}{c^2} \plus{} \frac {c^3}{a^2}\ge \frac {a^2}{b} \plus{} \frac {b^2}{c} \plus{} \frac {c^2}{a}$

Hey, This problem is from the VTRMC 2006. 3. Recall that the Fibonacci numbers $ F(n)$ are defined by $ F(0) \equal{} 0$, $ F(1) \equal{} 1$ and $ F(n) \equal{} F(n \minus{} 1) \plus{} F(n \minus{} 2)$ for $ n \geq 2$. Determine the last digit of $ F(2006)$ (e.g. the last digit of 2006 is 6). As, I and a friend were working on this we noticed an interesting relationship when writing the Fibonacci numbers in "mod" notation. Consider the following, 01 = 1 mod 10 01 = 1 mod 10 02 = 2 mod 10 03 = 3 mod 10 05 = 5 mod 10 08 = 6 mod 10 13 = 3 mod 10 21 = 1 mod 10 34 = 4 mod 10 55 = 5 mod 10 89 = 9 mod 10 Now, consider that between the first appearance and second apperance of $ 5 mod 10$, there is a difference of five terms. Following from this we see that the third appearance of $ 5 mod 10$ occurs at a difference 10 terms from the second appearance. Following this pattern we can create the following relationships. $ F(55) \equal{} F(05) \plus{} 5({2}^{2})$ This is pretty much as far as we got, any ideas?

Let $B$ be a subset of $\mathbb{Z}_{3}^{n}$ with the property that for every two distinct members $(a_{1},\ldots,a_{n})$ and $(b_{1},\ldots,b_{n})$ of $B$ there exist $1\leq i\leq n$ such that $a_{i}\equiv{b_{i}+1}\pmod{3}$. Prove that $|B| \leq 2^{n}$.

Several points are denoted on a white piece of paper. The distance between each two of the points is greater than $24$. A drop of ink was sprinkled over the paper covering an area smaller than $\pi$. Prove that there exists a vector $\overrightarrow v$ with $\overrightarrow v<1$, such that after translating all of the points by $v$ none of them is covered in ink.

We are given a set $S=\{z_1,z_2,\cdots ,z_{1993}\}$, where $z_1,z_2,\cdots ,z_{1993}$ are nonzero complex numbers (also viewed as nonzero vectors in the plane). Prove that we can divide $S$ into some groups such that the following conditions are satisfied: (1) Each element in $S$ belongs and only belongs to one group; (2) For any group $p$, if we use $T(p)$ to denote the sum of all memebers in $p$, then for any memeber $z_i (1\le i \le 1993)$ of $p$, the angle between $z_i$ and $T(p)$ does not exceed $90^{\circ}$; (3) For any two groups $p$ and $q$, the angle between $T(p)$ and $T(q)$ exceeds $90^{\circ}$ (use the notation introduced in (2)).

Find the maximum value of $f(x)=\int_0^1 t\sin (x+\pi t)\ dt$.

Given the vectors $ v_ {1}, \dots, v_ {n} $ and $ w_ {1}, \dots, w_ {n} $ in the plane with the following properties: for every $ 1 \leq i \leq n $ ,$ \left | v_{i} -w_{i} \right | \leq 1, $ and for every $ 1 \leq i <j \leq n $ ,$ \left | v_{i} -v_{j} \right | \ge 3 $ and $ v_{i} -w_ {i} \ne v_ {j} -w_ {j} $. Prove that for sets $ V = \left \{v_ {1}, \dots, v_{n } \right \} $ and $ W = \left \{w_ {1}, \dots, w_ {n} \right \}$, the set of $ V + (V \cup W) $ must have at least $ cn^{3/2} $ elements ,for some universal constant $ c>0 $ .

On a coordinate plane are placed four counters, each of whose centers has integer coordinates. One can displace any counter by the vector joining the centers of two of the other counters. Prove that any two preselected counters can be made to coincide by a finite sequence of moves. [i]Р. Sadykov[/i]

An object moves in two dimensions according to \[\vec{r}(t) = (4.0t^2 - 9.0)\vec{i} + (2.0t - 5.0)\vec{j}\] where $r$ is in meters and $t$ in seconds. When does the object cross the $x$-axis? $ \textbf{(A)}\ 0.0 \text{ s}\qquad\textbf{(B)}\ 0.4 \text{ s}\qquad\textbf{(C)}\ 0.6 \text{ s}\qquad\textbf{(D)}\ 1.5 \text{ s}\qquad\textbf{(E)}\ 2.5 \text{ s}$

Sequences $(x_n)_{n\ge1}$, $(y_n)_{n\ge1}$ satisfy the relations $x_n=4x_{n-1}+3y_{n-1}$ and $y_n=2x_{n-1}+3y_{n-1}$ for $n\ge1$. If $x_1=y_1=5$ find $x_n$ and $y_n$. Calculate $\lim_{n\rightarrow\infty}\frac{x_n}{y_n}$.

Square $ABCD$ has side length $68$. Let $E$ be the midpoint of segment $\overline{CD}$, and let $F$ be the point on segment $\overline{AB}$ a distance $17$ from point $A$. Point $G$ is on segment $\overline{EF}$ so that $\overline{EF}$ is perpendicular to segment $\overline{GD}$. The length of segment $\overline{BG}$ can be written as $m\sqrt{n}$ where $m$ and $n$ are positive integers, and $n$ is not divisible by the square of any prime. Find $m+n$.

Let P be an interior point of a regular n-gon $ A_1 A_2 ...A_n$, the lines $ A_i P$ meet the regular n-gon at another point $ B_i$, where $ i\equal{}1,2,...,n$. Prove that sums of all $ PA_i\geq$ sum of all $ PB_i$.