Found problems: 85335
Let $a$ and $ b$ be acute corners. Prove that if $\sin a$, $\sin b$, and $\sin (a + b)$ are rational numbers, then $\cos a$, $\cos b$, and $\cos (a + b)$ are also rational numbers.
The cricles $\Gamma_1$ and $\Gamma_2$ intersect in $ M $ and $N.$ A line that goes through $ M $ intersects the cricles $\Gamma_1$ and $\Gamma_2$ in $ A$ and $B$, such that $M\in(AB)$. The bisector of angle $ AMN $ intersects the circle $\Gamma_1$ in $D,$ and the bisector of angle $BMN$ intersects the circle $\Gamma_2$ in $C.$ Prove that the circle with diameter $CD$ splits the segment $AB$ in half.
Given an equilateral triangle $ABC$, let $M$ be an arbitrary point in space.
$(\text{a})$ Prove that one can construct a triangle from the segments $MA, MB, MC$.
$(\text{b})$ Suppose that $P$ and $Q$ are two points symmetric with respect to the center $O$ of $ABC$. Prove that the two triangles constructed from the segments $PA,PB,PC$ and $QA,QB,QC$ are of equal area.
$$\lim_{n \to \infty} nr\sqrt[2]{1-\cos \frac{2\pi}{n}}=?$$
A kingdom has $2021$ towns. All of the towns lie on a circle, and there is a one-way road going from every town to the next $101$ towns in a clockwise order. Each road is colored in one color. Additionally, it is known that for any ordered pair of towns $A$ and $B$ it is possible to find a path from $A$ to $B$ so that no two roads of the path would have the same color. Find the minimal number of road colors in the kingdom.
Functions $f_0, f_1,f_2,...$ are recursively defined by
$f_0(x) = x$ and $f_{2k+1} (x) = 3^{f_{2k}(x)}$ and $f_{2k+2} = 2^{f_{2k+1}(x)}$, $k = 0,1,2,...$ for all $x \in R$.
Find the greater one of the numbers $f_{10}(1)$ and $f_9(2)$.
Find all functions $f:\mathbb R\to \mathbb R$ such that \begin{align*} \left (x \left (f(x)-\dfrac{f(y)+f(z)}{2} \right) +y \left (f(y)-\dfrac{f(z)+f(x)}{2} \right ) +z\left (f(z)- \dfrac{f(x)+f(y)}{2} \right) \right )f(x+y+z)= \\ f(x^3)+f(y^3)+f(z^3)-3f(xyz) \end{align*} for all $x,y,z\in \mathbb R.$
For every integer $d \geq 1$, let $M_d$ be the set of all positive integers that cannot be written as a sum of an arithmetic progression with difference $d$, having at least two terms and consisting of positive integers. Let $A = M_1$, $B = M_2 \setminus \{2 \}, C = M_3$. Prove that every $c \in C$ may be written in a unique way as $c = ab$ with $a \in A, b \in B.$
What is the maximum possible score on this contest? Recall that on the NIMO 2013 Summer Contest, problems $1$, $2$, \dots, $15$ are worth $1$, $2$, \dots, $15$ points, respectively.
[i]Proposed by Evan Chen[/i]
Non-zero real numbers $a, b, c$ satisfy the condition $\frac{1}{a}+\frac{2}{b}+\frac{3}{c}= 0$.
Determine the value of $w =\frac{3b + 2c}{6a}+\frac{2c + 6a}{3b}+\frac{6a + 3b}{2c}$
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Find all the pairs of positive integers $(x,p)$ such that p is a prime, $x \leq 2p$ and $x^{p-1}$ is a divisor of $ (p-1)^{x}+1$.
Partition $\frac1{2002},\frac1{2003},\frac1{2004},\ldots,\frac{1}{2017}$ into two groups. Define $A$ the sum of the numbers in the first group, and $B$ the sum of the numbers in the second group. Find the partition such that $|A-B|$ attains it minimum and explains the reason.
Let be two real numbers $ a<b $ and a differentiable function $ f:[a,b]\longrightarrow\mathbb{R} $ that has a bounded derivative.
Show that if $ \frac{f(b)-f(a)}{b-a} $ is equal to the global supremum or infimum of $ f', $ then $ f $ is polynomial with degree $ 1. $
[i]Cătălin Țigăeru[/i]
Each cell of a $7\times7$ table is painted with one of several colours. It is known that for any two distinct rows the numbers of colours used to paint them are distinct and for any two distinct columns the numbers of colours used to paint them are distinct.What is the maximum possible number of colours in the table?
Sphere $S$ touches a plane. Let $A,B,C,D$ be four points on the plane such that no three of them are collinear. Consider the point $A'$ such that $S$ in tangent to the faces of tetrahedron $A'BCD$. Points $B',C',D'$ are defined similarly. Prove that $A',B',C',D'$ are coplanar and the plane $A'B'C'D'$ touches $S$.
[i]Proposed by Alexey Zaslavsky (Russia)[/i]
Show that for a triangle with radii of circumcircle and incircle equal to $ R$, $ r$ respectively, the inequality $ R \geq 2r$ holds.
There are $3$ students from Korea, China, and Japan, so total of $9$ students are present. How many ways are there to make them sit down in a circular table, with equally spaced and equal chairs, such that the students from the same country do not sit next to each other? If array $A$ can become array $B$ by rotation, these two arrays are considered equal.
Four lines are given in a plane so that any three of them determine a triangle. One of these lines is parallel to a median in the triangle determined by the other three lines. Prove that each of the other three lines also has this property.
The circumference inscribed on the triangle $ABC$ is tangent to the sides $BC$, $CA$ and $AB$ on the points $D$, $E$ and $F$, respectively. $AD$ intersect the circumference on the point $Q$. Show that the line $EQ$ meet the segment $AF$ at its midpoint if and only if $AC=BC$.
An infinite road has traffic lights at intervals of 1500m. The lights are all synchronised and are alternately green for $\frac 32$ minutes and red for 1 minute. For which $v$ can a car travel at a constant speed of $v$m/s without ever going through a red light?
Let $a,\,b,$ and $c$ be real numbers such that $a+b+c=\frac1{a}+\frac1{b}+\frac1{c}$ and $abc=5$. The value of $$\left(a-\frac1{b}\right)^3+\left(b-\frac1{c}\right)^3+\left(c-\frac1{a}\right)^3$$ can be written in the form $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $m+n$.
For a positive integer $n$ we denote by $s(n)$ the sum of the digits of $n$. Let $P(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$ be a polynomial, where $n \geqslant 2$ and $a_i$ is a positive integer for all $0 \leqslant i \leqslant n-1$. Could it be the case that, for all positive integers $k$, $s(k)$ and $s(P(k))$ have the same parity?
There are $n$ lamps and $k$ switches in a room. Initially, each lamp is either turned on or turned off. Each lamp is connected by a wire with $2020$ switches. Switching a switch changes the state of a lamp, that is connected to it, to the opposite state. It is known that one can switch the switches so that all lamps will be turned on. Prove, that it is possible to achieve the same result by switching the switches no more than $ \left \lfloor \dfrac{k}{2} \right \rfloor$ times.
[i]Proposed by T. Zimanov[/i]
Given an arbitrary triangle $ABC$, with $\angle A = 60^o$ and $AC < AB$. A circle with diameter $BC$, intersects $AB$ and $AC$ at $F$ and $E$, respectively. Lines $BE$ and $CF$ intersect at $D$. Let $\Gamma$ be the circumcircle of $BCD$, where the center of $\Gamma$ is $O$. Circle $\Gamma$ intersects the line $AB$ and the extension of $AC$ at $M$ and $N$, respectively. $MN$ intersects $BC$ at $P$. Prove that points $A$, $P$, $O$ lie on the same line.
Let $W$ be a regular octahedron and $O$ be its center. In a plane $P$ containing $O$ circles $k_1(O,r_1)$ and $k_2(O,r_2)$ are chosen so that $k_1 \subset P\cap W \subset k_2$. Prove that $\frac{r_1}{r_2}\le \frac{\sqrt3}{2}$