Found problems: 85335
Let $ ABC$ be a right angled triangle of area 1. Let $ A'B'C'$ be the points obtained by reflecting $ A,B,C$ respectively, in their opposite sides. Find the area of $ \triangle A'B'C'.$
Determine all positive real $M$ such that for any positive reals $a,b,c$, at least one of $a + \dfrac{M}{ab}, b + \dfrac{M}{bc}, c + \dfrac{M}{ca}$ is greater than or equal to $1+M$.
Let $p, q, r$ be prime numbers and $t, n$ be natural numbers such that $p^2 +qt =(p + t)^n$ and $p^2 + qr = t^4$ .
a) Show that $n < 3$.
b) Determine all the numbers $p, q, r, t, n$ that satisfy the given conditions.
Three digits are selected at random and without replacement from the set of digits $0$ through $9$. What is the probability that the three digits can be arranged to form a multiple of $5$?
$\text{(A) }\frac{17}{90}\qquad\text{(B) }\frac{7}{15}\qquad\text{(C) }\frac{1}{2}\qquad\text{(D) }\frac{8}{15}\qquad\text{(E) }\frac{1}{5}$
What is the sum of all $x$ that satisfy $|2x-4| = 2$?
$\textbf{(A) } 1\qquad\textbf{(B) } 2\qquad\textbf{(C) } 3\qquad\textbf{(D) } 4\qquad\textbf{(E) } 5$
$ABCD$ is a tetrahedron.
Let $K$ be the center of the incircle of $CBD$.
Let $M$ be the center of the incircle of $ABD$.
Let $L$ be the gravycenter of $DAC$.
Let $N$ be the gravycenter of $BAC$.
Suppose $AK$, $BL$, $CM$, $DN$ have one common point.
Is $ABCD$ necessarily regular?
A restaurant offers three desserts, and exactly twice as many appetizers as main courses. A dinner consists of an appetizer, a main course, and a dessert. What is the least number of main courses that the restaurant should offer so that a customer could have a different dinner each night in the year $ 2003$?
$ \textbf{(A)}\ 4 \qquad
\textbf{(B)}\ 5 \qquad
\textbf{(C)}\ 6 \qquad
\textbf{(D)}\ 7 \qquad
\textbf{(E)}\ 8$
Let $n\geq 3$ be an integer. Each vertex of a regular $n$-gon is labelled with a real number not exceeding $1$. For real numbers $a,b,c$ on any three consecutive vertices which are arranged clockwise in such an order, we have $c=|a-b|$. Determine the maximum value of the sum of all numbers in terms of $n$.
The degree measures of the angles of nondegenerate hexagon $ABCDEF$ are integers that form a non-constant arithmetic sequence in some order, and $\angle A$ is the smallest angle of the (not necessarily convex) hexagon. Compute the sum of all possible degree measures of $\angle A$.
[i]Proposed by Lewis Chen[/i]
Each vertex of a regular octagon is coloured either red or blue with equal probability. The probability that the octagon can then be rotated in such a way that all of the blue vertices end up at points that were originally red is $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$?
If $x$ and $y$ are positive integers such that $(x-4)(x-10)=2^y$, then Find maximum value of $x+y$
Kevin runs uphill at a speed that is $4$ meters per second slower than his speed when he runs downhill. Kevin takes a total of $80$ seconds to run up and down a hill on one path. Given that the path is $300$ meters long (he travels $600$ meters total), find how long Kevin takes to run up the hill in seconds.
Let $p > 3$ be a prime. Prove that if \[ \sum_{i=1 }^{p-1}{1\over i^p} = {n\over m}, \] with $\gdc(n,m) = 1$, then $p^3$ divides $n$.
Six teams participate in a hockey tournament. Each team plays exactly once against each other team. A team is awarded $3$ points for each game they win, $1$ point for each draw, and $0$ points for each game they lose. After the tournament, a ranking is made. There are no ties in the list. Moreover, it turns out that each team (except the very last team) has exactly $2$ points more than the team ranking one place lower.
Prove that the team that fi nished fourth won exactly two games.
4-Point $ P$ is taken on the segment $ BC$ of the scalene triangle $ ABC$ such that $ AP \neq AB,AP \neq AC$.$ l_1,l_2$ are the incenters of triangles $ ABP,ACP$ respectively. circles $ W_1,W_2$ are drawn centered at $ l_1,l_2$ and with radius equal to $ l_1P,l_2P$,respectively. $ W_1,W_2$ intersects at $ P$ and $ Q$. $ W_1$ intersects $ AB$ and $ BC$ at $ Y_1( \mbox{the intersection closer to B})$ and $ X_1$,respectively. $ W_2$ intersects $ AC$ and $ BC$ at $ Y_2(\mbox{the intersection closer to C})$ and $ X_2$,respectively.PROVE THE CONCURRENCY OF $ PQ,X_1Y_1,X_2Y_2$.
Find the product of all divisors of $1980^n$, $n \ge 1$.
Let $m,n$ and $a_1,a_2,\dots,a_m$ be arbitrary positive integers. Ali and Mohammad Play the following game. At each step, Ali chooses $b_1,b_2,\dots,b_m \in \mathbb{N}$ and then Mohammad chosses a positive integers $s$ and obtains a new sequence $\{c_i=a_i+b_{i+s}\}_{i=1}^m$, where $$b_{m+1}=b_1,\ b_{m+2}=b_2, \dots,\ b_{m+s}=b_s$$ The goal of Ali is to make all the numbers divisible by $n$ in a finite number of steps. FInd all positive integers $m$ and $n$ such that Ali has a winning strategy, no matter how the initial values $a_1, a_2,\dots,a_m$ are.
[hide=clarification] after we create the $c_i$ s, this sequence becomes the sequence that we continue playing on, as in it is our 'new' $a_i$[/hide]
Proposed by Shayan Gholami
Prove that the determinant of the matrix
$$\begin{pmatrix}
a_{1}^{2}+k & a_1 a_2 & a_1 a_3 &\ldots & a_1 a_n\\
a_2 a_1 & a_{2}^{2}+k & a_2 a_3 &\ldots & a_2 a_n\\
\ldots & \ldots & \ldots & \ldots & \ldots \\
a_n a_1& a_n a_2 & a_n a_3 & \ldots & a_{n}^{2}+k
\end{pmatrix}$$
is divisible by $k^{n-1}$ and find its other factor.
If $a,b,c$ are positive real numbers such that $a+b+c=abc$ prove that
$$\frac{abc}{3\sqrt{2}}\left ( \sum_{cyc}\frac{\sqrt{a^3+b^3}}{ab+1} \right )\geq \sum_{cyc}\frac{a}{a^2+1}$$
Six points are given so that there are not three on the same line and that the lengths of the segments determined by these points are all different. We consider all the triangles that they have their vertices at these points. Show that there is a segment that is both the shortest side of one of those triangles and the longest side of another.
Denote by $ H_n$ the linear space of $ n\times n$ self-adjoint complex matrices, and by $ P_n$ the cone of positive-semidefinite matrices in this space. Let us consider the usual inner product on $ H_n$
\[ \langle A,B\rangle \equal{} {\rm tr} AB\qquad (A,B\in H_n)\]
and its derived metric. Show that every $ \phi: P_n\to P_n$ isometry (that is a not necessarily surjective, distance preserving map with respect to the above metric) can be expressed as
\[ \phi(A) \equal{} UAU^* \plus{} X\qquad (A\in H_n)\]
or
\[ \phi(A) \equal{} UA^TU^* \plus{} X\qquad (A\in H_n)\]
where $ U$ is an $ n\times n$ unitary matrix, $ X$ is a positive-semidefinite matrix, and $ ^T$ and $ ^*$ denote taking the transpose and the adjoint, respectively.
Let $AL$ and $BK$ be angle bisectors in the non-isosceles triangle $ABC$ ($L$ lies on the side $BC$, $K$ lies on the side $AC$). The perpendicular bisector of $BK$ intersects the line $AL$ at point $M$. Point $N$ lies on the line $BK$ such that $LN$ is parallel to $MK$. Prove that $LN = NA$.
Let $ P$ be the product of all non-zero digits of the positive integer $ n$. For example, $ P(4) \equal{} 4$, $ P(50) \equal{} 5$, $ P(123) \equal{} 6$, $ P(2009) \equal{} 18$.
Find the value of the sum: P(1) + P(2) + ... + P(2008) + P(2009).
It is known that $4$ people $A, B, C$, and $D$ each have a $1/3$ probability of telling the truth. Suppose that
$\bullet$ $A$ makes a statement.
$\bullet$ $B$ makes a statement about the truthfulness of $A$’s statement.
$\bullet$ $C$ makes a statement about the truthfulness of $B$’s statement.
$\bullet$ $D$ says that $C$ says that $B$ says that $A$ was telling the truth.
What is the probability that $A$ was actually telling the truth?
For which positive integers $m$ does there exist an infinite arithmetic sequence of integers $a_1, a_2, . . .$ and an infinite geometric sequence of integers $g_1, g_2, . . .$ satisfying the following properties?
[list]
[*] $a_n - g_n$ is divisible by $m$ for all integers $n \ge 1$;
[*] $a_2 - a_1$ is not divisible by $m$.
[/list]
[i]Holden Mui[/i]