Found problems: 85335
2018 Brazil National Olympiad, 3
Let $ABC$ be an acute-angled triangle with circumcenter $O$ and orthocenter $H$. The circle with center $X_a$ passes in the points $A$ and $H$ and is tangent to the circumcircle of $ABC$. Define $X_b, X_c$ analogously, let $O_a, O_b, O_c$ the symmetric of $O$ to the sides $BC, AC$ and $AB$, respectively. Prove that the lines $O_aX_a, O_bX_b, O_cX_c$ are concurrents.
2009 Brazil Team Selection Test, 3
Let $P(x) = x^4 + ax^3 + bx^2 + cx + d$ be a monic polynomial of degree $4$. It is known that all the roots of $P$ are real, distinct and belong to the interval $[-1, 1]$.
(a) Prove that $P(x) > -4$ for all real $x$.
(b) Find the highest value of the real constant $k$ such that $P(x) > k$ for every real $x$ and for every polynomial $P(x)$ satisfying the given conditions.
2014 JHMMC 7 Contest, 12
Lev scores $91, 89, 88, 94, 87, 85$ on his first $6$ tests. After having his final exam, he (correctly) states that the average of all $7$ of his test scores is equal to his final exam score. What was Lev’s final exam score?
2007 Korea National Olympiad, 1
Consider the string of length $ 6$ composed of three characters $ a$, $ b$, $ c$. For each string, if two $ a$s are next to each other, or two $ b$s are next to each other, then replace $ aa$ by $ b$, and replace $ bb$ by $ a$. Also, if $ a$ and $ b$ are next to each other, or two $ c$s are next to each other, remove all two of them (i.e. delete $ ab$, $ ba$, $ cc$). Determine the number of strings that can be reduced to $ c$, the string of length 1, by the reducing processes mentioned above.
2024 Girls in Mathematics Tournament, 2
Show that there are no triples of positive integers $(x,y,z)$ satisfying the equation \[x^2= 5^y+3^z\]
2012 India IMO Training Camp, 3
In a $2\times n$ array we have positive reals s.t. the sum of the numbers in each of the $n$ columns is $1$. Show that we can select a number in each column s.t. the sum of the selected numbers in each row is at most $\frac{n+1}4$.
2014 Miklós Schweitzer, 8
Let $n\ge 1$ be a fixed integer. Calculate the distance $\inf_{p,f}\, \max_{0\le x\le 1} |f(x)-p(x)|$ , where $p$ runs over polynomials of degree less than $n$ with real coefficients and $f$ runs over functions $f(x)= \sum_{k=n}^{\infty} c_k x^k$ defined on the closed interval $[0,1]$ , where $c_k \ge 0$ and $\sum_{k=n}^{\infty} c_k=1$.
2010 Postal Coaching, 6
Let $a,b,c$ denote the sides of a triangle and $[ABC]$ the area of the triangle as usual.
$(a)$ If $6[ABC] = 2a^2+bc$, determine $A,B,C$.
$(b)$ For all triangles, prove that $3a^2+3b^2 - c^2 \ge 4 \sqrt{3} [ABC]$.
2005 MOP Homework, 7
Eight problems were given to each of $30$ students. After the test was given, point values of the problems were determined as follows: a problem is worth $n$ points if it is not solved by exactly $n$ contestants (no partial credit is given, only zero marks or full marks).
(a) Is it possible that the contestant having got more points that any other contestant had also solved less problems than any other contestant?
(b) Is it possible that the contestant having got less points than any other contestant has solved more problems than any other contestant?
2018 China Team Selection Test, 1
Let $p,q$ be positive reals with sum 1. Show that for any $n$-tuple of reals $(y_1,y_2,...,y_n)$, there exists an $n$-tuple of reals $(x_1,x_2,...,x_n)$ satisfying $$p\cdot \max\{x_i,x_{i+1}\} + q\cdot \min\{x_i,x_{i+1}\} = y_i$$ for all $i=1,2,...,2017$, where $x_{2018}=x_1$.
2018 Thailand TSTST, 3
Let $BC$ be a chord not passing through the center of a circle $\omega$. Point $A$ varies on the major arc $BC$. Let $E$ and $F$ be the projection of $B$ onto $AC$, and of $C$ onto $AB$ respectively. The tangents to the circumcircle of $\vartriangle AEF$ at $E, F$ intersect at $P$.
(a) Prove that $P$ is independent of the choice of $A$.
(b) Let $H$ be the orthocenter of $\vartriangle ABC$, and let $T$ be the intersection of $EF$ and $BC$. Prove that $TH \perp AP$.
2010 AMC 12/AHSME, 16
Positive integers $ a,b,$ and $ c$ are randomly and independently selected with replacement from the set $ \{ 1,2,3,\dots,2010 \}.$ What is the probability that $ abc \plus{} ab \plus{} a$ is divisible by $ 3$?
$ \textbf{(A)}\ \dfrac{1}{3} \qquad\textbf{(B)}\ \dfrac{29}{81} \qquad\textbf{(C)}\ \dfrac{31}{81} \qquad\textbf{(D)}\ \dfrac{11}{27} \qquad\textbf{(E)}\ \dfrac{13}{27}$
1991 Tournament Of Towns, (312) 2
$11$ girls and $n$ boys went for mushrooms. They have found $n^2+9n -2$ in total, and each child has found the same quantity. Which is greater: the number of girls or the number of boys?
(A. Tolpygo, Kiev)
2002 Romania National Olympiad, 2
Prove that any real number $0<x<1$ can be written as a difference of two positive and less than $1$ irrational numbers.
2015 Polish MO Finals, 2
Prove that diagonals of a convex quadrilateral are perpendicular if and only if inside of the quadrilateral there is a point, whose orthogonal projections on sides of the quadrilateral are vertices of a rectangle.
2021 Polish Junior MO First Round, 3
The numbers $a, b, c$ satisfy the condition $| a - b | = 2 | b - c | = 3 | c - a |$. Prove that $a = b = c$.
2024 Serbia National Math Olympiad, 6
Find all non-constant polynomials $P(x)$ with integer coefficients and positive leading coefficient, such that $P^{2mn}(m^2)+n^2$ is a perfect square for all positive integers $m, n$.
2020 Junior Balkаn MO, 3
Alice and Bob play the following game: Alice picks a set $A = \{1, 2, ..., n \}$ for some natural number $n \ge 2$. Then, starting from Bob, they alternatively choose one number from the set $A$, according to the following conditions: initially Bob chooses any number he wants, afterwards the number chosen at each step should be distinct from all the already chosen numbers and should differ by $1$ from an already chosen number. The game ends when all numbers from the set $A$ are chosen. Alice wins if the sum of all the numbers that she has chosen is composite. Otherwise Bob wins. Decide which player has a winning strategy.
Proposed by [i]Demetres Christofides, Cyprus[/i]
2004 Moldova Team Selection Test, 6
Find all functions $f:\mathbb R \to \mathbb R$ Such that for all real $x,y$:
$(x^2+xy+y^2)(f(x)-f(y))=f(x^3)-f(y^3)$
1999 Baltic Way, 4
For all positive real numbers $x$ and $y$ let
\[f(x,y)=\min\left( x,\frac{y}{x^2+y^2}\right) \]
Show that there exist $x_0$ and $y_0$ such that $f(x, y)\le f(x_0, y_0)$ for all positive $x$ and $y$, and find $f(x_0,y_0)$.
2018 Mathematical Talent Reward Programme, SAQ: P 4
Suppose $S$ is a finite subset of $\mathbb{R}$. If $f: S \rightarrow S$ is a function such that,
$$
\left|f\left(s_{1}\right)-f\left(s_{2}\right)\right| \leq \frac{1}{2}\left|s_{1}-s_{2}\right|, \forall s_{1}, s_{2} \in S
$$
Prove that, there exists a $x \in S$ such that $f(x)=x$
2022 Azerbaijan EGMO/CMO TST, N4
Let $n\geq 1$ be a positive integer. We say that an integer $k$ is a [i]fan [/i]of $n$ if $0\leq k\leq n-1$ and there exist integers $x,y,z\in\mathbb{Z}$ such that
\begin{align*}
x^2+y^2+z^2 &\equiv 0 \pmod n;\\
xyz &\equiv k \pmod n.
\end{align*}
Let $f(n)$ be the number of fans of $n$. Determine $f(2020)$.
2007 Princeton University Math Competition, 5
Find the values of $a$ such that $\log (ax+1) = \log (x-a) + \log (2-x)$ has a unique real solution.
2024 Belarus Team Selection Test, 2.3
A right triangle $ABC$ ($\angle A=90$) is inscribed in a circle $\omega$. Tangent to $\omega$ at $A$ intersects $BC$ at $P$, $B$ lies between $P$ and $C$. Let $M$ be the midpoint of the minor arc $AB$. $MP$ intersects $\omega$ at $Q$. Point $X$ lies on a ray $PA$ such that $\angle XCB=90$. Prove that line $XQ$ passes through the orthocenter of the triangle $ABO$
[i]Mayya Golitsyna[/i]
2017 AMC 8, 11
A square-shaped floor is covered with congruent square tiles. If the total number of tiles that lie on the two diagonals is 37, how many tiles cover the floor?
$\textbf{(A) }148\qquad\textbf{(B) }324\qquad\textbf{(C) }361\qquad\textbf{(D) }1296\qquad\textbf{(E) }1369$