Found problems: 85335
Let $ABC$ be an equilateral triangle, $M$ be a point inside it, and $A',B',C'$ be the intersections of $AM,\; BM,\; CM$ with the sides of $ABC$. If $A'',\; B'',\; C''$ are the midpoints of $BC$, $CA$, $AB$, show that there is a triangle with sides $A'A''$, $B'B''$ and $C'C''$.
[i]Laurențiu Panaitopol[/i]
How many prime numbers $ p $ the number $ p ^ 3-4 p + 9 $ is a perfect square
Find all pairs of natural numbers $(m, n)$ for which $2^m3^n +1$ is the square of some integer.
Compute the number of primes less than $40$ that are the sum of two primes.
Let $ABC$ be an acute triangle.A circle with diameter $BC$ meets $AB$ and $AC$ at $E$ and $F$,respectively. $M$ is midpoint of $BC$ and $P$ is point of intersection $AM$ with $EF$. $X$ is an arbitary point on arc $EF$ and $Y$ is the second intersection of $XP$ with a circle with diameter $BC$.Prove that $ \measuredangle XAY=\measuredangle XYM $.
Author:Ali zo'alam , Iran
On the side $AB$ of the non-isosceles triangle $ABC$, let the points $P$ and $Q$ be so that $AC = AP$ and $BC = BQ$. The perpendicular bisector of the segment $PQ$ intersects the angle bisector of the $\angle C$ at the point $R$ (inside the triangle). Prove that $\angle ACB + \angle PRQ = 180^o$.
Let $a$ and $b$ be fixed positive numbers. Find the smallest possible depending on $a$ and $b$ value of the sum
$$\frac{x^2}{(ay + bz)(az + by)}+\frac{y^2}{(az + bx)(ax + bz)}+\frac{z^2}{(ax + by)(ay + bx)},$$
where $x, y, z$ are positive real numbers.
Prove that there exist infinitely many pairs $ (a, b)$ of natural numbers not equal to $ 1$ such that $ b^b \plus{}a$ is divisible by $ a^a \plus{}b$.
Let $ I $ and $ 0 $ be the square identity and null matrices, both of size $ 2019 $. There is a square matrix $A$
with rational entries and size $ 2019 $ such that:
a) $ A ^ 3 + 6A ^ 2-2I = 0 $?
b) $ A ^ 4 + 6A ^ 3-2I = 0 $?
For $a,b,c$ positive reals, let \[N=\dfrac{a^2+b^2}{c^2+ab}+\dfrac{b^2+c^2}{a^2+bc}+\dfrac{c^2+a^2}{b^2+ca}.\] Find the minimum value of $\lfloor 2008N\rfloor$.
Find the greatest positive integer $n$ such that we can choose $2007$ different positive integers from $[2\cdot 10^{n-1},10^{n})$ such that for each two $1\leq i<j\leq n$ there exists a positive integer $\overline{a_{1}a_{2}\ldots a_{n}}$ from the chosen integers for which $a_{j}\geq a_{i}+2$.
[i]A. Ivanov, E. Kolev[/i]
The International Mathematical Olympiad is being organized in Japan, where a folklore belief is that the number $4$ brings bad luck. The opening ceremony takes place at the Grand Theatre where each row has the capacity of $55$ seats. What is the maximum number of contestants that can be seated in a single row with the restriction that no two of them are $4$ seats apart (so that bad luck during the competition is avoided)?
The trapezium ${ABCD}$, where ${AB}$ and ${CD}$ are parallel and ${AD < CD}$, is inscribed in the circle ${c}$. Let ${DP}$ be a chord of the circle, parallel to ${AC}$. Assume that the tangent to ${c}$ at ${D}$ meets the line ${AB}$ at ${E}$ and that ${PB}$ and ${DC}$ meet at ${Q}$. Show that ${EQ = AC}$.
Find all triples of integers $(x, y, z)$ such that $x^4 + 5y^4 = z^4$.
Let $N$ be the product of all odd primes less than $2^4$. What remainder does $N$ leave when divided by $2^4$?
$\text{(A) }5\qquad\text{(B) }7\qquad\text{(C) }9\qquad\text{(D) }11\qquad\text{(E) }13$
Compute all pairs of nonzero real numbers $(x,y)$ such that
$$\frac{x}{x^2+y}+\frac{y}{x+y^2}=-1\qquad\text{and}\qquad\frac{1}{x}+\frac{1}{y}=1.$$
Let $ABCM$ be a quadrilateral and $D$ be an interior point such that $ABCD$ is a parallelogram. It is known that $\angle AMB =\angle CMD$. Prove that $\angle MAD =\angle MCD$.
The function $f:\mathbb R^{\ge 0} \longrightarrow \mathbb R^{\ge 0}$ satisfies the following properties for all $a,b\in \mathbb R^{\ge 0}$:
[b]a)[/b] $f(a)=0 \Leftrightarrow a=0$
[b]b)[/b] $f(ab)=f(a)f(b)$
[b]c)[/b] $f(a+b)\le 2 \max \{f(a),f(b)\}$.
Prove that for all $a,b\in \mathbb R^{\ge 0}$ we have $f(a+b)\le f(a)+f(b)$.
[i]Proposed by Masoud Shafaei[/i]
Prove that if $p$ and $p+2$ are prime integers greater than 3, then 6 is a factor of $p+1$.
In a square lattice let us take a lattice triangle that has the smallest area among all the lattice triangles similar to it. Prove that the circumcenter of this triangle is not a lattice point.
At the beginning of school year in one of the first grade classes:
$i)$ every student had exatly $20$ acquaintances
$ii)$ every two students knowing each other had exactly $13$ mutual acquaintances
$iii)$ every two students not knowing each other had exactly $12$ mutual acquaintances
Find number of students in this class
A circle is inscribed in a triangle with sides $ 8$, $ 15$, and $ 17$. The radius of the circle is:
$ \textbf{(A)}\ 6 \qquad
\textbf{(B)}\ 2 \qquad
\textbf{(C)}\ 5 \qquad
\textbf{(D)}\ 3 \qquad
\textbf{(E)}\ 7$
$a)$ Let $n$ a positive integer. Prove that $gcd(n, \lfloor n\sqrt{2} \rfloor)<\sqrt[4]{8}\sqrt{n}$.
$b)$ Prove that there are infinitely many positive integers $n$ such that $gcd(n, \lfloor n\sqrt{2} \rfloor)>\sqrt[4]{7.99}\sqrt{n}$.
A piece of string is cut in two at a point selected at random. The probability that the longer piece is at least $x$ times as large as the shorter piece is
$\textbf{(A) }\frac{1}{2}\qquad\textbf{(B) }\frac{2}{x}\qquad\textbf{(C) }\frac{1}{x+1}\qquad\textbf{(D) }\frac{1}{x}\qquad \textbf{(E) }\frac{2}{x+1}$
Consider the set $A = \{0, 1, 2, . . . , 2^{2n} - 1\}$. The function $f : A \rightarrow A$ is given by: $f(x_0 + 2x_1 + 2^2x_2 + ... + 2^{2n-1}x_{2n-1})=$$(1 - x_0) + 2x_1 + 2^2(1 - x_2) + 2^3x_3 + ... + 2^{2n-1}x_{2n-1}$
for every $0-1$ sequence $(x_0, x_1, . . . , x_{2n-1})$. Show that if $a_1, a_2, . . . , a_9$ are consecutive terms of an arithmetic progression, then the sequence $f(a_1), f(a_2), . . . , f(a_9)$ is not increasing.