Found problems: 85335
Prove that for any prime number $p > 3$ exist integers $x, y, k$ that meet conditions: $0 < 2k < p$ and $kp + 3 = x^2 + y^2$.
Let $Q$ be an $n$-by-$n$ real orthogonal matrix, and let $u\in \mathbb{R}^n$ be a unit column vector (that is, $u^Tu=1$). Let $P=I-2uu^T$, where $I$ is the $n$-by-$n$ identity matrix. Show that if $1$ is not an eigenvalue of $Q$, then $1$ is an eigenvalue of $PQ$.
Let $P$ be an arbitrary point on the side $BC$ of triangle $ABC$ and let $D$ be the tangency point between the incircle of the triangle $ABC$ and the side $BC$. If $Q$ and $R$ are respectively the incenters in the triangles $ABP$ and $ACP$, prove that $\angle QDR$ is a right angle.
Prove that the triangle $QDR$ is isosceles if and only if $P$ is the foot of the altitude from $A$ in the triangle $ABC$.
Prove that there is a function $ F:\mathbb{N}\longrightarrow\mathbb{N} $ satisfying $ (F\circ F) (n) =n^2, $ for all $ n\in\mathbb{N} . $
$ 1 \le n \le 455$ and $ n^3 \equiv 1 \pmod {455}$. The number of solutions is ?
$\textbf{(A)}\ 9 \qquad\textbf{(B)}\ 6 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 1 \qquad\textbf{(E)}\ \text{None}$
The function $f$ is defined for non-negative integers and satisfies the condition $f(n) = f(f(n + 11))$, if $n \le 1999$ and
$f(n) = n - 5$, if $n > 1999$. Find all solutions of the equation $f(n) = 1999$.
Determine the number of $ 8$-tuples of nonnegative integers $ (a_1,a_2,a_3,a_4,b_1,b_2,b_3,b_4)$ satisfying $ 0\le a_k\le k$, for each $ k \equal{} 1,2,3,4$, and $ a_1 \plus{} a_2 \plus{} a_3 \plus{} a_4 \plus{} 2b_1 \plus{} 3b_2 \plus{} 4b_3 \plus{} 5b_4 \equal{} 19$.
Given a permutation $\sigma = (a_1,a_2,a_3,...a_n)$ of $(1,2,3,...n)$ , an ordered pair $(a_j,a_k)$ is called an inversion of $\sigma$ if $a \leq j < k \leq n$ and $a_j > a_k$. Let $m(\sigma)$ denote the no. of inversions of the permutation $\sigma$. Find the average of $m(\sigma)$ as $\sigma$ varies over all permutations.
Determine the largest prime which divides both $2^{24}-1$ and $2^{16}-1$.
[i]2017 CCA Math Bonanza Individual Round #6[/i]
A company has a secret safe box that is locked by six locks. Several copies of the keys are distributed among the directors of the company. Each key can unlock exactly one lock. Each director has three keys for three different locks. No two directors can unlock the same three locks. No two directors together can unlock the safe. What is the maximum possible number of directors in the company?
In an acute triangle $AB$ the heights $ BE$ and $CF$ intersect at the orthocenter $H$, and $M$ is the midpoint of $BC$. The line $EF$ intersects the lines $MH$ and $BC$ at the points $P$ and $T$ , respectively. $AP$ intersects the cirumcscribed circle of $\vartriangle ABC$ for second time at the point $Q$ . Prove that $\angle AQT= 90^o$.
(Fedir Yudin)
Can a countably infinite set have an uncountable collection of non-empty subsets such that the intersection of any two of them is finite?
Let $n$ be a positive integer relatively prime to $6$. We paint the vertices of a regular $n$-gon with three colours so that there is an odd number of vertices of each colour. Show that there exists an isosceles triangle whose three vertices are of different colours.
Determine all triples of real numbers $(a,b,c)$ that satisfy simultaneously the equations:
$$\begin{cases} a(b^2 + c) = c(c + ab) \\ b(c^2 + a) = a(a + bc) \\ c(a^2 + b) = b(b + ca) \end{cases}$$
$\boxed{A4}$Let $m_1,m_2,m_3,n_1,n_2$ and $n_3$ be positive real numbers such that
\[(m_1-n_1)(m_2-n_2)(m_3-n_3)=m_1m_2m_3-n_1n_2n_3\]
Prove that
\[(m_1+n_1)(m_2+n_2)(m_3+n_3)\geq8m_1m_2m_3\]
Consider a regular $2020$-gon circumscribed into a circle of radius $ 1$. Given three vertices of this polygon such that they form an isosceles triangle, let $X$ be the expected area of the isosceles triangle they create. $X$ can be written as $\frac{1}{m \tan((2\pi)/n)}$ where $m$ and $n$ are integers. Compute $m + n$.
Harry has $3$ sisters and $5$ brothers. His sister Harriet has $X$ sisters and $Y$ brothers. What is the product of $X$ and $Y$?
$ \text{(A)}\ 8\qquad\text{(B)}\ 10\qquad\text{(C)}\ 12\qquad\text{(D)}\ 15\qquad\text{(E)}\ 18 $
Suppose $x,y,z\in \mathbb R^+$. Prove that \[\frac {x}{\sqrt{yz+4xy+4xz}}+\frac {y}{\sqrt{zx+4yz+4yx}}+\frac {z}{\sqrt{xy+4zx+4zy}}\geq 1\].
Let $P(x) = x^3 - 6x^2 - 5x + 4$. Suppose that $y$ and $z$ are real numbers such that
\[ zP(y) = P(y - n) + P(y + n) \]
for all reals $n$. Evaluate $P(y)$.
Let $n>3$ be a positive integer. Equilateral triangle ABC is divided into $n^2$ smaller congruent equilateral triangles (with sides parallel to its sides). Let $m$ be the number of rhombuses that contain two small equilateral triangles and $d$ the number of rhombuses that contain eight small equilateral triangles. Find the difference $m-d$ in terms of $n$.
Let $n$ be a positive integer, and Megavan has a $(3n+1)\times (3n+1)$ board. All squares, except one, are tiled by non-overlapping $1\times 3$ triominoes. In each step, he can choose a triomino that is untouched in the step right before it, and then shift this triomino horizontally or vertically by one square, as long as the triominoes remain non-overlapping after this move.
Show that there exist some $k$, such that after $k$ moves Megavan can no longer make any valid moves irregardless of the initial configuration, and find the smallest possible $k$ for each $n$.
[i](Note: While he cannot undo a move immediately before the current step, he may still choose to move a triomino that has already been moved at least two steps before.)[/i]
[i]Proposed by Ivan Chan Kai Chin[/i]
Let $ ABC$ be a triangle with $ \angle ABC\equal{}\angle ACB\equal{}70^{\circ}$. Let point $ D$ on side $ BC$ such that $ AD$ is the altitude, point $ E$ on side $ AB$ such that $ \angle ACE\equal{}10^{\circ}$, and point $ F$ is the intersection of $ AD$ and $ CE$. Prove that $ CF\equal{}BC$.
Point $O$ is the center of the circumcircle of the acute triangle $ABC$. The line $AO$ intersects the side $BC$ at point $D$ so that $OD = BD = 1/3 BC$ . Find the angles of the triangle $ABC$. Justify the answer.
Two lines parallel to the $x$-axis cut the graph of $y=ax^3+bx^2+cx+d$ in points $A,C,E$ and $B,D,F$ respectively, in that order from left to right. Prove that the length of the projection of the segment $CD$ onto the $x$-axis equals the sum of the lengths of the projections of $AB$ and $EF$.
Let $ABC$ be a acute, non-isosceles triangle. $D,\ E,\ F$ are the midpoints of sides $AB,\ BC,\ AC$, resp. Denote by $(O),\ (O')$ the circumcircle and Euler circle of $ABC$. An arbitrary point $P$ lies inside triangle $DEF$ and $DP,\ EP,\ FP$ intersect $(O')$ at $D',\ E',\ F'$, resp. Point $A'$ is the point such that $D'$ is the midpoint of $AA'$. Points $B',\ C'$ are defined similarly.
a. Prove that if $PO=PO'$ then $O\in(A'B'C')$;
b. Point $A'$ is mirrored by $OD$, its image is $X$. $Y,\ Z$ are created in the same manner. $H$ is the orthocenter of $ABC$ and $XH,\ YH,\ ZH$ intersect $BC, AC, AB$ at $M,\ N,\ L$ resp. Prove that $M,\ N,\ L$ are collinear.