Found problems: 85335
Let $ A,B\in\mathcal{M} \left( \mathbb{R} \right) $ that satisfy $ AB=O_3. $ Prove that:
[b]a)[/b] The function $ f:\mathbb{C}\longrightarrow\mathbb{C} $ defined as $ f(x)=\det \left( A^2+B^2+xBA \right) $ is a polynomial one, of degree at most $ 2. $
[b]b)[/b] $ \det\left( A^2+B^2 \right)\ge 0. $
An equilateral polygon has unit side length and alternating interior angle measures of $15^o$ and $300^o$. Compute the area of this polygon.
Let $\vartriangle ABC$ be acute with orthocenter $H$. Let $X$ be a point on $BC$ such that $B-X-C$. Let $\Gamma$ be the circumscribed circle of $\vartriangle BHX$ and $\Gamma_2$ be the circumscribed circle of $\vartriangle CHX$. Let $E$ be the intersection of $AB$ with $\Gamma$ , and $D$ be the intersection of $AC$ with $\Gamma_2$. Let $L$ be the intersection of line $HD$ with $\Gamma$ and $J$ be the intersection of line $EH$ with $\Gamma_2$. Prove that points $L$, $X$, and $J$ are collinear.
Notation: $A-B-C$ means than points $A,B,C$ are collinear in that order i.e. $ B$ lies between $ A$ and $C$.
Prove that there exist distinct natural numbers $m_1,m_2, \cdots , m_k$ satisfying the conditions
\[\pi^{-1984}<25-\left(\frac{1}{m_1}+\frac{1}{m_2}+\cdots+\frac{1}{m_k}\right)<\pi^{-1960}\]
where $\pi$ is the ratio between a circle and its diameter.
Let $ABCD$ be a convex quadrilateral inscribed in a circle of radius $1$. Prove that \[ 0< (AB+BC+CD+AD)-(AC+BD) < 4. \]
For every positive integer $n$, let $S (n)$ be the sum of the digits of $n$. Find, if any, a $171$-digit positive integer $n$ such that $7$ divides $S (n)$ and $7$ divides $S (n + 1)$.
Let $f(x) = \frac{1}{1-\frac{3x}{16}}$. Consider the sequence $\{ 0, f(0), f(f(0)), f^3(0), \dots \}$ Find the smallest $L$ such that $f^n(0) \leq L$ for all $n$. If the sequence is unbounded, write none as your answer.
A designer took a wooden cube $5 \times 5 \times 5$, divided each face into unit squares and painted each square black, white or red so that any two squares with a common side have different colours. What is the least possible number of black squares? (Squares with a common side may belong to the same face of the cube or to two different faces.)
[i](8 points)[/i]
[i]Mikhail Evdokimov[/i]
How many positive integers less than 4000 are not divisible by 2, not divisible by 3, not divisible by 5, and not divisible by 7?
What is \[\frac{2+4+6}{1+3+5} - \frac{1+3+5}{2+4+6}?\]
$ \textbf{(A)}\ -1 \qquad
\textbf{(B)}\ \frac{5}{36} \qquad
\textbf{(C)}\ \frac{7}{12} \qquad
\textbf{(D)}\ \frac{147}{60} \qquad
\textbf{(E)}\ \frac{43}{3} $
Let $n$ be an integer greater than $3$. Prove that all the roots of the polynomial
$P(x) = x^n - 5x^{n-1} + 12x^{n-2}- 15x^{n-3} + a_{n-4}x^{n-4} +...+ a_0$ cannot be both real and positive.
Vika calls some positive integers [i]nice[/i], and it is known that among any ten consecutive positive integers there is at least one nice. Prove that there are infinitely many positive integers $n$ for which $ab-cd=2n^2$ for some pairwise distinct nice numbers $a,b,c,$ and $d$
How many four-digit integers $abcd$, with $a \neq 0$, have the property that the three two-digit integers $ab<bc<cd$ form an increasing arithmetic sequence? One such number is $4692$, where $a=4$, $b=6$, $c=9$, and $d=2$.
$\textbf{(A)}\ 9\qquad\textbf{(B)}\ 15\qquad\textbf{(C)}\ 16\qquad\textbf{(D)}\ 17\qquad\textbf{(E)}\ 20$
Consider a tree with $n$ vertices, labeled with $1,\ldots,n$ in a way that no label is used twice. We change the labeling in the following way - each time we pick an edge that hasn't been picked before and swap the labels of its endpoints. After performing this action $n-1$ times, we get another tree with its labeling a permutation of the first graph's labeling.
Prove that this permutation contains exactly one cycle.
Let $ a,b,c$ be positive numbers, satisfying $ abc\geq 1$. Prove that
\[ a^{3} \plus{} b^{3} \plus{} c^{3} \geq ab \plus{} bc \plus{} ca.\]
Show that the edges of a connected simple (no loops and no multiple edges) finite graph can be oriented so that the number of edges leaving each vertex is even if and only if the total number of edges is even
In a triangle $ABC$, let $H, I$ and $O$ be the orthocentre, incentre and circumcentre, respectively. If the points $B, H, I, C$ lie on a circle, what is the magnitude of $\angle BOC$ in degrees?
$1993$ points are arranged in a circle. At time $0$ each point is arbitrarily labeled $+1$ or $-1$. At times $n = 1, 2, 3, ...$ the vertices are relabeled. At time $n$ a vertex is given the label $+1$ if its two neighbours had the same label at time $n-1$, and it is given the label $-1$ if its two neighbours had different labels at time $n-1$. Show that for some time $n > 1$ the labeling will be the same as at time $1.$
For every positive integer $ n$, $ b(n)$ denote the number of positive divisors of $ n$ and $ p(n)$ denote the sum of all positive divisors of $ n$. For example, $ b(14)\equal{}4$ and $ p(14)\equal{}24$. Let $ k$ be a positive integer greater than $ 1$.
(a) Prove that there are infinitely many positive integers $ n$ which satisfy $ b(n)\equal{}k^2\minus{}k\plus{}1$.
(b) Prove that there are finitely many positive integers $ n$ which satisfy $ p(n)\equal{}k^2\minus{}k\plus{}1$.
Four different positive integers less than 10 are chosen randomly. What is the probability that their sum is odd?
Prove that every positive integer can be written as a finite sum of distinct integral powers of the golden ratio.
Let $ABC$ be a triangle with $\angle A=\angle C=30^{\circ}.$ Points $D,E,F$ are chosen on the sides $AB,BC,CA$ respectively so that $\angle BFD=\angle BFE=60^{\circ}.$ Let $p$ and $p_1$ be the perimeters of the triangles $ABC$ and $DEF$, respectively. Prove that $p\le 2p_1.$
Given a quadrilateral $ ABCD $ inscribed in a circle. The images of the points $ A $ and $ C $ in symmetry with respect to the line $ BD $ are the points $ A' $ and $ C' $, respectively, and the images of the points $ B $ and $ D $ in symmetry with respect to the line $ AC $ are the points $ B'$ and $D'$ respectively. Prove that the points $ A' $, $ B' $, $ C' $, $ D' $ lie on the circle.
A configuration of several checkers at the centers of squares on a rectangular sheet of grid paper is called [i]boring [/i] if some four checkers occupy the vertices of a rectangle with sides parallel to those of the sheet.
(a) Prove that any configuration of more than $3mn/4$ checkers on an $m\times n$ grid is boring.
(b) Prove that any configuration of $26$ checkers on a $7\times 7$ grid is boring.
In obtuse triangle $ABC$ where $\angle B > 90^\circ$ let $H$ and $O$ be its orthocenter and circumcenter respectively. Let $D$ be the foot of the altitude from $A$ to $HC$ and $E$ be the foot of the altitude from $B$ to $AC$ such that $O,E,D$ lie on a line. If $OC=8$ and $OE=4$, find the area of triangle $HAB$.
[i]Proposed by Harry Kim[/i]