Found problems: 85335
1977 AMC 12/AHSME, 29
Find the smallest integer $n$ such that \[(x^2+y^2+z^2)^2\le n(x^4+y^4+z^4)\] for all real numbers $x,y,$ and $z$.
$\textbf{(A) }2\qquad\textbf{(B) }3\qquad\textbf{(C) }4\qquad\textbf{(D) }6\qquad \textbf{(E) }\text{There is no such integer }n.$
2009 Sharygin Geometry Olympiad, 2
A cyclic quadrilateral is divided into four quadrilaterals by two lines passing through its inner point. Three of these quadrilaterals are cyclic with equal circumradii. Prove that the fourth part also is cyclic quadrilateral and its circumradius is the same.
(A.Blinkov)
2010 Malaysia National Olympiad, 7
A line segment of length 1 is given on the plane. Show that a line segment of length $\sqrt{2010}$ can be constructed using only a straightedge and a compass.
1966 German National Olympiad, 6
Prove the following theorem:
If the intersection of any plane that has more than one point in common with the surface $F$ is a circle, then $F$ is a sphere (surface).
2001 Korea Junior Math Olympiad, 7
Finite set $\{a_1, a_2, ..., a_n, b_1, b_2, ..., b_n\}=\{1, 2, …, 2n\}$ is given. If $a_1<a_2<...<a_n$ and $b_1>b_2>...>b_n$, show that
$$\sum_{i=1}^n |a_i-b_i|=n^2$$
1968 Spain Mathematical Olympiad, 5
Find the locus of the center of a rectangle, whose four vertices lies on the sides of a given triangle.
VI Soros Olympiad 1999 - 2000 (Russia), 11.2
Let $$f(x) = (...((x - 2)^2 - 2)^2 - 2)^2... - 2)^2$$
(here there are $n$ brackets $( )$). Find $f''(0)$
2003 SNSB Admission, 3
Let be the set $ \Lambda = \left\{ \lambda\in\text{Hol} \left[ \mathbb{C}\longrightarrow\mathbb{C} \right] |z\in\mathbb{C}\implies |\lambda (z)|\le e^{|\text{Im}(z)|} \right\} . $ Show that:
$ \text{(1)}\sin\in\Lambda $
$ \text{(2)}\sum_{p\in\mathbb{Z}}\frac{1}{(1+2p)^2} =\frac{\pi^2}{4} $
$ \text{(3)} f\in\Lambda\implies \left| f'(0) \right|\le 1 $
II Soros Olympiad 1995 - 96 (Russia), 10.1
Find the smallest $a$ for which the equation $x^2-ax +21 = 0$ has a root that is a natural number.
2010 Federal Competition For Advanced Students, P2, 5
Two decompositions of a square into three rectangles are called substantially different, if reordering the rectangles does not change one into the other.
How many substantially different decompositions of a $2010 \times 2010$ square in three rectangles with integer side lengths are there such that the area of one rectangle is equal to the arithmetic mean of the areas of the other rectangles?
2013 Online Math Open Problems, 23
Let $ABCDE$ be a regular pentagon, and let $F$ be a point on $\overline{AB}$ with $\angle CDF=55^\circ$. Suppose $\overline{FC}$ and $\overline{BE}$ meet at $G$, and select $H$ on the extension of $\overline{CE}$ past $E$ such that $\angle DHE=\angle FDG$. Find the measure of $\angle GHD$, in degrees.
[i]Proposed by David Stoner[/i]
1953 Putnam, B3
Solve the equations
$$ \frac{dy}{dx}=z(y+z)^n, \;\; \; \frac{dz}{dx} = y(y+z)^n,$$
given the initial conditions $y=1$ and $z=0$ when $x=0.$
1978 USAMO, 4
(a) Prove that if the six dihedral (i.e. angles between pairs of faces) of a given tetrahedron are congruent, then the tetrahedron is regular.
(b) Is a tetrahedron necessarily regular if five dihedral angles are congruent?
2020 CCA Math Bonanza, I14
An ant starts at the point $(0,0)$ in the coordinate plane. It can make moves from lattice point $(x_1,y_1)$ to lattice point $(x_2,y_2)$ whenever $x_2\geq x_1$, $y_2\geq y_1$, and $(x_1,y_1)\neq(x_2,y_2)$. For all nonnegative integers $m,n$, define $a_{m,n}$ to be the number of possible sequences of moves from $(0,0)$ to $(m,n)$ (e.g. $a_{0,0}=1$ and $a_{1,1}=3$). Compute
\[
\sum_{m=0}^{\infty}\sum_{n=0}^{\infty} \frac{a_{m,n}}{10^{m+n}}.
\]
[i]2020 CCA Math Bonanza Individual Round #14[/i]
2017 All-Russian Olympiad, 3
There are $100$ dwarfes with weight $1,2,...,100$. They sit on the left riverside. They can not swim, but they have one boat with capacity 100. River has strong river flow, so every dwarf has power only for one passage from right side to left as oarsman. On every passage can be only one oarsman. Can all dwarfes get to right riverside?
1977 IMO Longlists, 60
Suppose $x_0, x_1, \ldots , x_n$ are integers and $x_0 > x_1 > \cdots > x_n.$ Prove that at least one of the numbers $|F(x_0)|, |F(x_1)|, |F(x_2)|, \ldots, |F(x_n)|,$ where
\[F(x) = x^n + a_1x^{n-1} + \cdots+ a_n, \quad a_i \in \mathbb R, \quad i = 1, \ldots , n,\]
is greater than $\frac{n!}{2^n}.$
2003 China Team Selection Test, 2
Let $S$ be a finite set. $f$ is a function defined on the subset-group $2^S$ of set $S$. $f$ is called $\textsl{monotonic decreasing}$ if when $X \subseteq Y\subseteq S$, then $f(X) \geq f(Y)$ holds. Prove that: $f(X \cup Y)+f(X \cap Y ) \leq f(X)+ f(Y)$ for $X, Y \subseteq S$ if and only if $g(X)=f(X \cup \{ a \}) - f(X)$ is a $\textsl{monotonic decreasing}$ funnction on the subset-group $2^{S \setminus \{a\}}$ of set $S \setminus \{a\}$ for any $a \in S$.
1996 Cono Sur Olympiad, 2
Consider a sequence of real numbers defined by:
$a_{n + 1} = a_n + \frac{1}{a_n}$ for $n = 0, 1, 2, ...$
Prove that, for any positive real number $a_0$, is true that $a_{1996}$ is greater than $63$.
2011 LMT, 18
Let $x$ and $y$ be distinct positive integers below $15$. For any two distinct numbers $a, b$ from the set $\{2, x,y\}$, $ab + 1$ is always a positive square. Find all possible values of the square $xy + 1$.
2018 Romania National Olympiad, 2
In the square $ABCD$ the point $E$ is located on the side $[AB]$, and $F$ is the foot of the perpendicular from $B$ on the line $DE$. The point $L$ belongs to the line $DE$, such that $F$ is between $E$ and $L$, and $FL = BF$. $N$ and $P$ are symmetric of the points $A , F$ with respect to the lines $DE, BL$, respectively. Prove that:
a) The quadrilateral $BFLP$ is square and the quadrilateral $ALND$ is rhombus.
b) The area of the rhombus $ALND$ is equal to the difference between the areas of the squares $ABCD$ and $BFLP$.
1991 Swedish Mathematical Competition, 2
$x, y$ are positive reals such that $x - \sqrt{x} \le y - 1/4 \le x + \sqrt{x}$. Show that $y - \sqrt{y} \le x - 1/4 \le y + \sqrt{y}$.
1973 AMC 12/AHSME, 9
In $ \triangle ABC$ with right angle at $ C$, altitude $ CH$ and median $ CM$ trisect the right angle. If the area of $ \triangle CHM$ is $ K$, then the area of $ \triangle ABC$ is
$ \textbf{(A)}\ 6K \qquad
\textbf{(B)}\ 4\sqrt3\ K \qquad
\textbf{(C)}\ 3\sqrt3\ K \qquad
\textbf{(D)}\ 3K \qquad
\textbf{(E)}\ 4K$
the 12th XMO, Problem 2
Let $a_1,a_2,\cdots,a_{22}\in [1,2],$ find the maximum value of
$$\dfrac{\sum\limits_{i=1}^{22}a_ia_{i+1}}{\left( \sum\limits_{i=1}^{22}a_i\right) ^2}$$where $a_{23}=a_1.$
1962 All-Soviet Union Olympiad, 7
Let $a;b;c;d>0$ such that $abcd=1$. Prove that $a^2+b^2+c^2+d^2+a(b+c)+b(c+d)+c(d+a)\ge 10$
2009 CIIM, Problem 1
Prove that for any positive integer $n$ the number $\left(\frac{3+\sqrt{17}}{2}\right)^n+\left(\frac{3-\sqrt{17}}{2}\right)^n $ is an odd integer.