Given a set of $n>2$ planar vectors. A vector from this set is called [i]long[/i], if its length is not less than the length of the sum of other vectors in this set. Prove that if each vector is long, then the sum of all vectors equals to zero. [i]N. Agakhanov[/i]

Let $ n $ be a natural number. How many numbers of the form $ \pm 1\pm 2\pm 3\pm\cdots\pm n $ are there?

Let $\mathbb Z$ be the set of integers, and let $f\colon \mathbb Z \to \mathbb Z$ be a function. Prove that there are infinitely many integers $c$ such that the function $g\colon \mathbb Z \to \mathbb Z$ defined by $g(x) = f(x) + cx$ is not bijective. [i]Note: [/i] A function $g\colon \mathbb Z \to \mathbb Z$ is bijective if for every integer $b$, there exists exactly one integer $a$ such that $g(a) = b$.

Find all functions $g:\mathbb{N}\rightarrow\mathbb{N}$ such that \[\left(g(m)+n\right)\left(g(n)+m\right)\] is a perfect square for all $m,n\in\mathbb{N}.$ [i]Proposed by Gabriel Carroll, USA[/i]

How many different positive integers divide $10!$ ?

Points $M$ and $N$ are considered in the interior of triangle $ABC$ such that $\angle MAB = \angle NAC$ and $\angle MBA = \angle NBC$. Prove that $$\frac{AM \cdot AN}{AB \cdot AC}+ \frac{BM\cdot BN}{BA \cdot BC}+ \frac{CM \cdot CN }{CA \cdot CB}=1$$

For every positive $a, b,c, d$ such that $a + c \le ac$ and $b + d \le bd$ prove that $\frac{ab}{a + b} +\frac{bc}{b + c} +\frac{cd}{c + d} +\frac{da}{d + a} \ge 4$

A table with three rows and 100 columns is given. Initially, in the left cell of each row there are $400\cdot 3^{100}$ chips. At one move, Petya marks some (but at least one) chips on the table, and then Vasya chooses one of the three rows. After that, all marked chips in the selected row are shifted to the right by a cell, and all marked chips in the other rows are removed from the table. Petya wins if one of the chips goes beyond the right edge of the table; Vasya wins if all the chips are removed. Who has a winning strategy? [i]Proposed by P. Svyatokum, A. Khuzieva and D. Shabanov[/i]

Prove for every positive integer $n{:}$ \[ \frac {1 \cdot 3 \cdots (2n - 1)}{2 \cdot 4 \cdots (2n)} < \frac 1{\sqrt{3n}}\]

Let triangle $A_1BC$ have sides $A_1B=5$, $A_1C=12$, and $BC=13$. For all natural numbers $i$, let $B_i$ be the foot of the altitude from $A_i$ to $BC$, let $A_{2i}$ be the foot of the altitude from $B_i$ to $A_1B$, and let $A_{2i+1}$ be the foot of the altitude from $B_i$ to $A_1C$. \[ \sum_{i=1}^{7}A_iB_i = \frac{p}{q}\] Find $p+q$. [i]2022 CCA Math Bonanza Individual Round #13[/i]

Let $A_n$ be the set of all sequences with length $n$ and members of the set $\{1,2…q\}$. We denote with $B_n$ a subset of $A_n$ with a minimal number of elements with the following property: For each sequence $a_1,a_2,...,a_n$ from $A_n$ there exist a sequence $b_1,b_2,...,b_n$ from $B_n$ such that $a_i\neq b_i$ for each $i=1,2,....,n$. Prove that, if $q>n$, then $|B_n |=n+1$.

Determine for which $ n$ positive integer the equation: $ a \plus{} b \plus{} c \plus{} d \equal{} n \sqrt {abcd}$ has positive integer solutions.

Prove that if the roots of the equation $x^2 + px + q = 0$ are real, then for any real number $a$ the roots of the equation $$x^2 + px + q + (x + a) (2x + p) = 0$$ are also real.

Let $a,b,c$ be given positive integers. Prove that there exists some positive integer $N$ such that \[ a\mid Nbc+b+c,\ b\mid Nca+c+a,\ c\mid Nab+a+b \] if and only if, denoting $d=\gcd(a,b,c)$ and $a=dx$, $b=dy$, $c=dz$, the positive integers $x,y,z$ are pairwise coprime, and also $\gcd(d,xyz) \mid x+y+z$. (Dan Schwarz)

Let $ABC$ be an acute triangle with orthocenter $H$ and circumcenter $O$. Let $E$ and $F$ be points on the line segments $AC$ and $AB$ respectively such that $AEHF$ is a parallelogram. Prove that $\vert OE \vert = \vert OF \vert$.

In isosceles trapezoid $ABCD$ with $AB < CD$ and $BC = AD$, the angle bisectors of $\angle A$ and $\angle B$ intersect $CD$ at $E$ and $F$ respectively, and intersect each other outside the trapezoid at $G$. Given that $AD = 8$, $EF = 3$, and $EG = 4$, the area of $ABCD$ can be expressed as $\frac{a\sqrt{b}}{c}$ for positive integers $a, b$, and $c$, with $a$ and $c$ relatively prime and $b$ squarefree. Find $10000a +100b +c$.

On the exterior of a non-equilateral triangle $ABC$ consider the similar triangles $ABM,BCN$ and $CAP$, such that the triangle $MNP$ is equilateral. Find the angles of the triangles $ABM,BCN$ and $CAP$. [i]Nicolae Bourbacut[/i]

Suppose that $P(x)$ is a monic cubic polynomial with integer roots, and suppose that $\frac{P(a)}{a}$ is an integer for exactly $6$ integer values of $a$. Suppose furthermore that exactly one of the distinct numbers $\frac{P(1) + P(-1)}{2}$ and $\frac{P(1) - P(-1)}{2}$ is a perfect square. Given that $P(0) > 0$, find the second-smallest possible value of $P(0).$ [i]Proposed by Andrew Wu[/i]

Let a,b,c Postive real numbers such that $a+b+c\geq 6$. Find the minimum value $A=\sum_{cyc}{a^2}$+$\sum_{cyc}{\frac{a}{b^2+c+1}}$

Let $ABC$ be a triangle with $AB \neq AC$ and circumcenter $O$. The bisector of $\angle BAC$ intersects $BC$ at $D$. Let $E$ be the reflection of $D$ with respect to the midpoint of $BC$. The lines through $D$ and $E$ perpendicular to $BC$ intersect the lines $AO$ and $AD$ at $X$ and $Y$ respectively. Prove that the quadrilateral $BXCY$ is cyclic.

Petya and Vasya live in neighboring houses (see the plan in the figure). Vasya lives in the fourth entrance. It is known that Petya runs to Vasya by the shortest route (it is not necessary walking along the sides of the cells) and it does not matter from which side he runs around his house. Determine in which entrance he lives Petya . [img]https://cdn.artofproblemsolving.com/attachments/b/1/741120341a54527b179e95680aaf1c4b98ff84.png[/img]

What is the number of structural isomers with the molecular formula $\ce{C6H14}$? $ \textbf{(A) }\text{Three} \qquad\textbf{(B) }\text{Four} \qquad\textbf{(C) }\text{Five} \qquad\textbf{(D) }\text{Six}\qquad$

Suppose that $ \mathcal F$ is a family of subsets of $ X$. $ A,B$ are two subsets of $ X$ s.t. each element of $ \mathcal{F}$ has non-empty intersection with $ A, B$. We know that no subset of $ X$ with $ n \minus{} 1$ elements has this property. Prove that there is a representation $ A,B$ in the form $ A \equal{} \{a_1,\dots,a_n\}$ and $ B \equal{} \{b_1,\dots,b_n\}$, such that for each $ 1\leq i\leq n$, there is an element of $ \mathcal F$ containing both $ a_i, b_i$.

Prove that we can find an infinite set of positive integers of the from $2^n-3$ (where $n$ is a positive integer) every pair of which are relatively prime.

Given triangle $ PQR$ with $ \overline{RS}$ bisecting $ \angle R$, $ PQ$ extended to $ D$ and $ \angle n$ a right angle, then: [asy]path anglemark2(pair A, pair B, pair C, real t=8, bool flip=false) { pair M,N; path mark; M=t*0.03*unit(A-B)+B; N=t*0.03*unit(C-B)+B; if(flip) mark=Arc(B,t*0.03,degrees(C-B)-360,degrees(A-B)); else mark=Arc(B,t*0.03,degrees(A-B),degrees(C-B)); return mark; } unitsize(1.5cm); defaultpen(linewidth(.8pt)+fontsize(8pt)); pair P=(0,0), R=(3,2), Q=(4,0); pair S0=bisectorpoint(P,R,Q); pair Sp=extension(P,Q,S0,R); pair D0=bisectorpoint(R,Sp), Np=midpoint(R--Sp); pair D=extension(Np,D0,P,Q), M=extension(Np,D0,P,R); draw(P--R--Q); draw(R--Sp); draw(P--D--M); draw(anglemark2(Sp,P,R,17)); label("$p$",P+(0.35,0.1)); draw(anglemark2(R,Q,P,11)); label("$q$",Q+(-0.17,0.1)); draw(anglemark2(R,Np,D,8,true)); label("$n$",Np+(+0.12,0.07)); draw(anglemark2(R,M,D,13,true)); label("$m$",M+(+0.25,0.03)); draw(anglemark2(M,D,P,29)); label("$d$",D+(-0.75,0.095)); pen f=fontsize(10pt); label("$R$",R,N,f); label("$P$",P,S,f); label("$S$",Sp,S,f); label("$Q$",Q,S,f); label("$D$",D,S,f);[/asy]$ \textbf{(A)}\ \angle m \equal{} \frac {1}{2}(\angle p \minus{} \angle q) \qquad \textbf{(B)}\ \angle m \equal{} \frac {1}{2}(\angle p \plus{} \angle q)$ $ \textbf{(C)}\ \angle d \equal{} \frac {1}{2} (\angle q \plus{} \angle p) \qquad \textbf{(D)}\ \angle d \equal{} \frac {1}{2}\angle m \qquad \textbf{(E)}\ \text{none of these is correct}$