Found problems: 85335
may be challenge for beginner section, but anyone is able to solve it if you really try.
show that for every natural $n > 1$ we have: $(n-1)^2|\ n^{n-1}-1$
Let $a,b,c$ be three positive real numbers such that $ a \le b \le c$ and $a+b+c=1$. Prove that
\[\frac{a+c}{\sqrt{a^2+c^2}}+\frac{b+c}{\sqrt{b^2+c^2}}+\frac{a+b}{\sqrt{a^2+b^2}} \le \frac{3\sqrt{6}(b+c)^2}{\sqrt{(a^2+b^2)(b^2+c^2)(c^2+a^2)}}.\]
[i]Owen Goff.[/i]
In a triangle $\triangle ABC$ it is given that $(\sin A+\sin B):(\sin B+\sin C):(\sin C+\sin A)=9:10:11$.
Find the value of $480\cos A$
Consider $\triangle ABC$ and a point $M$ in its interior so that $\angle MAB = 10^\circ$, $\angle MBA = 20^\circ$, $\angle MCA = 30^\circ$ and $\angle MAC = 40^\circ$. What is $\angle MBC$?
Find the number of subsets of $\{1, 2, \cdots, 2000 \}$, the sum of whose elements is divisible by $5$.
Let $ a,b,c$ be real numbers. Prove that $ (ab\plus{}bc\plus{}ca\minus{}1)^2 \le (a^2\plus{}1)(b^2\plus{}1)(c^2\plus{}1)$.
Find all positive integers $x, y, z$ that satisfy the conditions: $$[x,y,z] =(x,y)+(y,z) + (z,x), x\le y\le z, (x,y,z) = 1$$
The symbols $[m,n]$ and $(m,n)$ respectively represent positive integers, the least common multiple and the greatest common divisor of $m$ and $n$.
We have triangulated a convex $(n+1)$-gon $P_0P_1\dots P_n$ (i.e., divided it into $n-1$ triangles with $n-2$ non-intersecting diagonals). Prove that the resulting triangles can be labelled with the numbers $1,2,\dots,n-1$ such that for any $i\in\{1,2,\dots,n-1\}$, $P_i$ is a vertex of the triangle with label $i$.
A truck is initially moving at velocity $v$. The driver presses the brake in order to slow the truck to a stop. The brake applies a constant force $F$ to the truck. The truck rolls a distance $x$ before coming to a stop, and the time it takes to stop is $t$.
Which of the following expressions is equal the initial momentum of the truck (i.e. the momentum before the driver starts braking)?
$\textbf{(A) } Fx\\
\textbf{(B) } Ft/2\\
\textbf{(C) } Fxt\\
\textbf{(D) } 2Ft\\
\textbf{(E) } 2Fx/v$
Isabella is making sushi. She slices a piece of salmon into the shape of a solid triangular prism. The prism is $2$ cm thick, and its triangular faces have side lengths $7$ cm, $ 24$cm, and $25$ cm. Find the volume of this piece of salmon in cm$^3$.
[i]Proposed by Isabella Li[/i]
Prove that there exist no positive integers $m$ and $n$ such that $m > 5$ and $(m - 1)! + 1 = m^n$.
Cut a square into eight acute-angled triangles.
Let $\alpha,\beta,\gamma$ be angles of a triangle. Two of them can be expressed using an auxiliary angle $\varphi$ in a way that $$\alpha=\varphi+\frac\pi4,\quad\beta=\pi-3\varphi.$$ Show that $\alpha>\gamma.$
Find the maximum number of rectangles with sides equal to 1 and 2 and parallel to the coordinate axes
such that each two have an area equal to 1 in common.
Let $n>100$ be a positive integer and originally the number $1$ is written on the blackboard. Petya and Vasya play the following game: every minute Petya represents the number of the board as a sum of two distinct positive fractions with coprime nominator and denominator and Vasya chooses which one to delete. Show that Petya can play in such a manner, that after $n$ moves, the denominator of the fraction left on the board is at most $2^n+50$, no matter how Vasya acts.
Let $A,B,C,D$ be the vertices of a rhombus, let $k_1$ be the circle through $B,C$ and $D$, let $k_2$ be the circle through $A,C$ and $D$, let $k_3$ be the circle through $A,B$ and $D$, let $k_4$ be the circle through $A,B$ and $C$. Prove that the tangents to $k_1$ and $k_3$ at $B$ form the same angle as the tangents to $k_2$ and $k_4$ at $A$.
Let $(x_{n}) \ n\geq 1$ be a sequence of real numbers with $x_{1}=1$ satisfying $2x_{n+1}=3x_{n}+\sqrt{5x_{n}^{2}-4}$
a) Prove that the sequence consists only of natural numbers.
b) Check if there are terms of the sequence divisible by $2011$.
Find all integers $x$ and $y$ such that $x^3\pm y^3 =2001p$, where $p$ is prime.
If $a, b, c$ are real numbers such that $a+b+c=6$ and $ab+bc+ca = 9$, find the sum of all possible values of the expression $\lfloor a \rfloor + \lfloor b \rfloor + \lfloor c \rfloor$.
Find all functions $f : R \to R$ such that $x f(x)-y f(y) = (x-y)f(x+y)$ for all $x,y \in R$.
Let $ n $ be a positive integer and $ n $ real numbers $ a_1,a_2,\ldots ,a_n $ such that $ a_1^2+a_2^2+\cdots +a_n^2=1. $ Show that $$ \sum_{1\le ij\le n} a_ia_j<2\sqrt n. $$
[i]Russian math competition[/i]
Let $r, s,$ and $t$ be the distinct roots of $x^3- 2022x^2 + 2022x + 2022.$ Compute $$\frac{1}{1-r^2} + \frac{1}{1-s^2} + \frac{1}{1-t^2}.$$
Find all integer numbers $x$ and $y$ such that:
\[(y^3+xy-1)(x^2+x-y)=(x^3-xy+1)(y^2+x-y).\]
[i]Proposed by Mahyar Sefidgaran[/i]
[b]a)[/b] Let $ \left( x_n \right)_{n\ge 1} $ be a sequence of real numbers having the property that $ \left| x_{n+1} -x_n \right|\leqslant 1/2^n, $ for any $ n\geqslant 1. $
Show that $ \left( x_n \right)_{n\ge 1} $ is convergent.
[b]b)[/b] Create a sequence $ \left( y_n \right)_{n\ge 1} $ of real numbers that has the following properties:
$ \text{(i) } \lim_{n\to\infty } \left( y_{n+1} -y_n \right) = 0 $
$ \text{(ii) } $ is bounded
$ \text{(iii) } $ is divergent
[i]Eugen Popa[/i]
The square $0\le x\le 1$, $0\le y\le 1$ is drawn in the plane $Oxy$. A grasshopper sitting at a point $M$ with noninteger coordinates outside this square jumps to a new point which is symmetrical to $M$ with respect to the leftmost (from the grasshopper’s point of view) vertex of the square. Prove that no matter how many times the grasshopper jumps, it will never reach the distance more than $10 d$ from the center $C$ of the square, where $d$ is the distance between the initial position $M$ and the center $C$.
(A Kanel)