Found problems: 85335
2024 District Olympiad, P2
Let $ABC$ be a triangle inscribed in the circle $\mathcal{C}(O,1)$. Denote by $s(M)=OH_1^2+OH_2^2+OH_3^2,$ $(\forall) M \in\mathcal{C}\setminus \left\{A,B,C\right\},$ where $H_1,H_2,H_3,$ are the orthocenters of the triangles $MAB,~MBC$ and $MCA.$
$a)$ Prove that if $ABC$ is equilateral$,$ then $s(M)=6,(\forall) M \in\mathcal{C}\setminus \left\{A,B,C\right\},$
$b)$ Prove that if there exist three distinct points $M_1,M_2,M_3\in\mathcal{C}\setminus \left\{A,B,C\right\}$ such that $s(M_1)=$$s(M_2)$$=s(M_3),$ then $ABC$ is equilateral$.$
2017 Saint Petersburg Mathematical Olympiad, 2
A circle passing through vertices $A$ and $B$ of triangle $ABC$ intersects the sides $AC$ and $BC$ again at points $P$ and $Q$, respectively. Given that the median from vertex $C$ bisect the arc $PQ$ of the circle. Prove that $ABC$ is an isosceles triangle.
2004 AMC 8, 6
After Sally takes 20 shots, she has made $55\%$ of her shots. After she takes 5 more shots, she raises her percentage to $56\%$. How many of the last 5 shots did she make?
$\textbf{(A)} 1 \qquad\textbf{(B)} 2 \qquad\textbf{(C)} 3 \qquad\textbf{(D)} 4 \qquad\textbf{(E)} 5$
2020 China Second Round Olympiad, 2
Let $n\geq3$ be a given integer, and let $a_1,a_2,\cdots,a_{2n},b_1,b_2,\cdots,b_{2n}$ be $4n$ nonnegative reals, such that $$a_1+a_2+\cdots+a_{2n}=b_1+b_2+\cdots+b_{2n}>0,$$ and for any $i=1,2,\cdots,2n,$ $a_ia_{i+2}\geq b_i+b_{i+1},$ where $a_{2n+1}=a_1,$ $a_{2n+2}=a_2,$ $b_{2n+1}=b_1.$ Detemine the minimum of $a_1+a_2+\cdots+a_{2n}.$
2006 District Olympiad, 1
Let $x,y,z$ be positive real numbers. Prove the following inequality: \[ \frac 1{x^2+yz} + \frac 1{y^2+zx } + \frac 1{z^2+xy} \leq \frac 12 \left( \frac 1{xy} + \frac 1{yz} + \frac 1{zx} \right). \]
2016 China Western Mathematical Olympiad, 2
Let $\astrosun O_1$ and $\astrosun O_2$ intersect at $P$ and $Q$, their common external tangent touches $\astrosun O_1$ and $\astrosun O_2$ at $A$ and $B$ respectively. A circle $\Gamma$ passing through $A$ and $B$ intersects $\astrosun O_1$, $\astrosun O_2$ at $D$, $C$. Prove that $\displaystyle \frac{CP}{CQ}=\frac{DP}{DQ}$
1983 Austrian-Polish Competition, 5
Let $a_1 < a_2 < a_3 < a_4$ be given positive numbers. Find all real values of parameter $c$ for which the system
$$\begin{cases} x_1 + x_2 + x_3 + x_4 = 1 \\
a_1x_1 + a_2 x_2 + a_3x_3 + a_4 x_4 = c \\
a_1^2x_1 + a_2^2 x_2 + a_3^2x_3 + a_4^2 x_4 = c^2\end{cases}$$
has a solution in nonnegative $(x_1,x_2,x_3,x_4)$ real numbers.
2013 District Olympiad, 3
On the sides $(AB)$ and $(AC)$ of the triangle $ABC$ are considered the points $M$ and $N$ respectively so that $ \angle ABC =\angle ANM$. Point $D$ is symmetric of point $A$ with respect to $B$, and $P$ and $Q$ are the midpoints of the segments $[MN]$ and $[CD]$, respectively. Prove that the points $A, P$ and $Q$ are collinear if and only if $AC = AB \sqrt {2}$
2001 Romania National Olympiad, 3
Let $f:\mathbb{R}\rightarrow[0,\infty )$ be a function with the property that $|f(x)-f(y)|\le |x-y|$ for every $x,y\in\mathbb{R}$.
Show that:
a) If $\lim_{n\rightarrow \infty} f(x+n)=\infty$ for every $x\in\mathbb{R}$, then $\lim_{x\rightarrow\infty}=\infty$.
b) If $\lim_{n\rightarrow \infty} f(x+n)=\alpha ,\alpha\in[0,\infty )$ for every $x\in\mathbb{R}$, then $\lim_{x\rightarrow\infty}=\alpha$.
2001 Romania National Olympiad, 2
We consider a matrix $A\in M_n(\textbf{C})$ with rank $r$, where $n\ge 2$ and $1\le r\le n-1$.
a) Show that there exist $B\in M_{n,r}(\textbf{C}), C\in M_{r,n}(\textbf{C})$, with $%Error. "rank" is a bad command.
B=%Error. "rank" is a bad command.
C = r$, such that $A=BC$.
b) Show that the matrix $A$ verifies a polynomial equation of degree $r+1$, with complex coefficients.
2013 Federal Competition For Advanced Students, Part 1, 2
Solve the following system of equations in rational numbers:
\[ (x^2+1)^3=y+1,\\ (y^2+1)^3=z+1,\\ (z^2+1)^3=x+1.\]
VII Soros Olympiad 2000 - 01, 10.4
An acute-angled triangle is inscribed in a circle of radius $R$. The distance between the center of the circle and the point of intersection of the medians of the triangle is $d$. Find the radius of a circle inscribed in a triangle whose vertices are the feet of the altitudes of this triangle.
2019 Argentina National Olympiad Level 2, 1
We say that three positive integers $a$, $b$ and $c$ form a [i]family[/i] if the following two conditions are satisfied:
[list]
[*]$a + b + c = 900$.
[*]There exists an integer $n$, with $n \geqslant 2$, such that $$\frac{a}{n-1}=\frac{b}{n}=\frac{c}{n+1}.$$
[/list]
Determine the number of such families.
2011 Swedish Mathematical Competition, 1
Determine all positive integers $k$, $\ell$, $m$ and $n$, such that $$\frac{1}{k!}+\frac{1}{\ell!}+\frac{1}{m!} =\frac{1}{n!} $$
2020 Belarusian National Olympiad, 11.8
$10$ teams participated in a football tournament: every two teams played each other exactly once. After the end of the tournament it turned out that all teams got different amount of points and some teams won more games, than the winner of the tournament, call them strong.
What is the maximum number of teams that could be strong? (In football the winner of the match gets three points, the loser - 0 points, and if the match ends in a draw both teams get 1 point.)
2005 Gheorghe Vranceanu, 3
$ \lim_{n\to\infty }\left( \frac{1}{e}\sum_{i=0}^n \frac{1}{i!} \right)^{n!} $
2022/2023 Tournament of Towns, P2
Consider two coprime integers $p{}$ and $q{}$ which are greater than $1{}$ and differ from each other by more than $1{}$. Prove that there exists a positive integer $n{}$ such that \[\text{lcm}(p+n, q+n)<\text{lcm}(p,q).\]
2009 Bosnia And Herzegovina - Regional Olympiad, 2
Let $ABC$ be an equilateral triangle such that length of its altitude is $1$. Circle with center on the same side of line $AB$ as point $C$ and radius $1$ touches side $AB$. Circle rolls on the side $AB$. While the circle is rolling, it constantly intersects sides $AC$ and $BC$. Prove that length of an arc of the circle, which lies inside the triangle, is constant
2020 Mexico National Olympiad, 5
A four-element set $\{a, b, c, d\}$ of positive integers is called [i]good[/i] if there are two of them such that their product is a mutiple of the greatest common divisor of the remaining two. For example, the set $\{2, 4, 6, 8\}$ is good since the greatest common divisor of $2$ and $6$ is $2$, and it divides $4\times 8=32$.
Find the greatest possible value of $n$, such that any four-element set with elements less than or equal to $n$ is good.
[i]Proposed by Victor and Isaías de la Fuente[/i]
2014 Contests, 2
Given the rational numbers $r$, $q$, and $n$, such that $\displaystyle\frac1{r+qn}+\frac1{q+rn}=\frac1{r+q}$, prove that $\displaystyle\sqrt{\frac{n-3}{n+1}}$ is a rational number.
1962 Putnam, B5
Prove that for every integer $n$ greater than $1:$
$$\frac{3n+1}{2n+2} < \left( \frac{1}{n} \right)^{n} + \left( \frac{2}{n} \right)^{n}+ \ldots+\left( \frac{n}{n} \right)^{n} <2.$$
Durer Math Competition CD Finals - geometry, 2018.C+1
Prove that you can select two adjacent sides of any quadrilateral and supplement them in order to create a parallelogram, the resulting parallelogram contains the original quadrilateral .
1989 National High School Mathematics League, 10
A positive number, if its fractional part, integeral part, and itself are geometric series, then the number is________.
2013 IFYM, Sozopol, 2
The point $P$, from the plane in which $\Delta ABC$ lies, is such that if $A_1,B_1$, and $C_1$ are the orthogonal projections of $P$ on the respective altitudes of $ABC$, then $AA_1=BB_1=CC_1=t$. Determine the locus of $P$ and length of $t$.
III Soros Olympiad 1996 - 97 (Russia), 9.6
In triangle $ABC$, angle $B$ is not right. The circle inscribed in $ABC$ touches $AB$ and $BC$ at points $C_1$ and $A_1$, and the feet of the altitudes drawn to the sides $AB$ and $BC$ are points $C_2$ and $A_2$. Prove that the intersection point of the altitudes of triangle $A_1BC_1$ is the center of the circle inscribed in triangle $A_2BC_2$.