Found problems: 85335
An isosceles trapezoid $ABCD$ ($AB = CD$) is given. A point $P$ on its circumcircle is such that segments $CP$ and $AD$ meet at point $Q$. Let $L$ be tha midpoint of$ QD$. Prove that the diagonal of the trapezoid is not greater than the sum of distances from the midpoints of the lateral sides to ana arbitrary point of line $PL$.
Let $k\ge 3$ be an integer. Prove that if $n>\binom k3$, then for any $3n$ pairwise different real numbers $a_i,b_i,c_i$ ($1\le i\le n$), among the numbers $a_i+b_i$, $a_i+c_i$, $b_i+c_i$, one can find at least $k+1$ pairwise different numbers. Show that this is not always the case when $n=\binom k3$.
Prove that there are infinitely many n’s for which there exists a partition of $\{1,2,...,3n\}$ into subsets $\{a_1,...,a_n\}, \{b_1,...,b_n\}, \{c_1,...,c_n\}$ such that $a_i +b_i = c_i$ for all $i$, and prove that there are infinitely many $n$’s for which there is no such partition.
Show that a path on a rectangular grid which starts at the northwest corner, goes through each point on the grid exactly once, and ends at the southeast corner divides the grid into two equal halves:
(a) those regions opening north or east; and
(b) those regions opening south or west.
[img]https://cdn.artofproblemsolving.com/attachments/b/e/aa20c9f9bc44bd1e5a9b9e86d49debf0f821b7.png[/img]
(The figure above shows a path meeting the conditions of the problem on a $5 \times 8$ grid.
The shaded regions are those opening north or east while the rest open south or west.)
Let $n$ be a positive integer. A sequence $a_1$, $a_2$,$ ...$ , $a_n$ is called [i]good [/i] if the following conditions hold:
$\bullet$ For each $i \in \{1, 2, ..., n\}$, $1 \le a_i \le n$
$\bullet$ For all positive integers $i, j$ with $1 \le i \le j \le n$, the expression $a_i + a_{i+1} + ...+ a_j$ is not divisible by $ n + 1$.
Find the number of good sequences (in terms of $n$).
Fuzzy likes isosceles trapezoids. He can choose lengths from $1, 2, \dots, 8$, where he may choose any amount of each length. He takes a multiset of three integers from $1, \dots, 8$. From this multiset, one length will become a base length, one will become a diagonal length, and one will become a leg length. He uses each element as either a diagonal, leg, or base length exactly once. Fuzzy is happy if he can use these lengths to make an isosceles trapezoid such that the undecided base has nonzero rational length. How many multiset choices can he make? (Multisets are unordered)
[i]Proposed by Timothy Qian[/i]
For certain real constants $ p, q, r$, we are given a system of equations
$$\begin{cases} a^2 + b + c = p \\
a + b^2 + c = q \\
a + b + c^2 = r \end{cases}$$
What is the maximum number of solutions of real triplets $(a, b, c)$ across all possible $p, q, r$? Give an example of the $p$, $q$, $r$ that achieves this maximum.
Let $n$ be a positive integer, and set $S$ be the set of all integers in $\{1,2,\dots,n\}$ which are relatively prime to $n$.
Set $S_1 = S \cap \left(0, \frac n3 \right]$, $S_2 = S \cap \left( \frac n3, \frac {2n}3 \right]$, $S_3 = S \cap \left( \frac{2n}{3}, n \right]$.
If the cardinality of $S$ is a multiple of $3$, prove that $S_1$, $S_2$, $S_3$ have the same cardinality.
Suppose that $ x_1, x_2, x_3, \ldots$ are positive real numbers for which \[ x^n_n \equal{} \sum^{n\minus{}1}_{j\equal{}0} x^j_n\] for $ n \equal{} 1, 2, 3, \ldots$ Prove that $ \forall n,$ \[ 2 \minus{} \frac{1}{2^{n\minus{}1}} \leq x_n < 2 \minus{} \frac{1}{2^n}.\]
$f(x) = x^3+3x^2 - 1$. Find the number of real solutions of $f(f(x)) = 0$.
On sides $BC,CA,AB$ of a triangle $ABC$ points $D,E,F$ respectively are chosen so that $AD,BE,CF$ have a common point, say $G$. Suppose that one can inscribe circles in the quadrilaterals $AFGE,BDGF,CEGD$ so that each two of them have a common point. Prove that triangle $ABC$ is equilateral.
One day, the temperature increases steadily from a low of $ 45^\circ \text{F}$ in the early morning to a high of $ 70^\circ \text{F}$ in the late afternoon. At how many times from early morning to late afternoon was the temperature an integer in both Fahrenheit and Celsius? Recall that $ C \equal{} \frac {5}{9}(F \minus{} 32)$.
Find all pairs $(x,n)$ of positive integers for which $x^n + 2^n + 1$ divides $x^{n+1} +2^{n+1} +1$.
The vertices of a regular nonagon are colored such that $1)$ adjacent vertices are different colors and $2)$ if $3$ vertices form an equilateral triangle, they are all different colors. Let $m$ be the minimum number of colors needed for a valid coloring, and n be the total number of colorings using $m$ colors. Determine $mn$. (Assume each vertex is distinguishable.)
[i](6 pts)[/i] Find all positive integers $n \ge 3$ such that there exists a permutation $a_{1}, a_{2}, \dots, a_{n}$ of $1, 2, \dots, n$ such that $a_{1}, 2a_{2}, \dots, na_{n}$ can be rearranged into an arithmetic progression.
Prove that the inequality \[ \frac{ x+y}{x^2-xy+y^2 } \leq \frac{ 2\sqrt 2 }{\sqrt{ x^2 +y^2 } } \] holds for all real numbers $x$ and $y$, not both equal to 0.
Let $ABC$ be a triangle and $D, E$ and $F$ the foots of heights from $A, B$ and $C$ respectively. Let $D_1$ be such a point on $EF$, that $DF = D_1 E$ where $E$ is between $D_1$ and $F$. Similarly, let $D_2$ be such a point on $EF$, that $DE = D_2 F$ where $F$ is between $E$ and $D_2$. Let the bisector of $DD_1$ intersect $AB$ at $P$ and let the bisector of $DD_2$ intersect $AC$ at $Q$.
Prove that, $PQ$ bisects $BC$.
Let $P(x)=x^6-x^5-x^3-x^2-x$ and $a,b,c$ and $d$ be the roots of the equation $x^4-x^3-x^2-1=0$, then determine the value of $P(a)+P(b)+P(c)+P(d)$
(A)$5$
(B)$6$
(C)$7$
(D)$8$
Let $ABC$ be a triangle and $D$ is a point inside of the triangle, such that $\angle DBC=60^{\circ}$ and $\angle DCB=\angle DAB=30^{\circ}$. Let $M$ and $N$ be the midpoints of $AC$ and $BC$, respectively. Prove that $\angle DMN=90^{\circ}$.
In scalene $\triangle ABC$, the tangent from the foot of the bisector of $\angle A$ to the incircle of $\triangle ABC$, other than the line $BC$, meets the incircle at point $K_a$. Points $K_b$ and $K_c$ are analogously defined. Prove that the lines connecting $K_a$, $K_b$, $K_c$ with the midpoints of $BC$, $CA$, $AB$, respectively, have a common point on the incircle.
Let $D$ be a point on the side $BC$ of $\triangle ABC$. If $AB=8$, $AC=7$, $BD=2$, and $CD=1$, find $AD$.
What is the largest integer $n$ for which $\frac{2008!}{31^n}$ is an integer?
The vertices of a square sheet of paper are $ A$, $ B$, $ C$, $ D$. The sheet is folded in a way that the point $ D$ is mapped to the point $ D'$ on the side $ BC$. Let $ A'$ be the image of $ A$ after the folding, and let $ E$ be the intersection point of $ AB$ and $ A'D'$. Let $ r$ be the inradius of the triangle $ EBD'$. Prove that $ r\equal{}A'E$.
Denote by $ M$ midpoint of side $ BC$ in an isosceles triangle $ \triangle ABC$ with $ AC = AB$. Take a point $ X$ on a smaller arc $ \overarc{MA}$ of circumcircle of triangle $ \triangle ABM$. Denote by $ T$ point inside of angle $ BMA$ such that $ \angle TMX = 90$ and $ TX = BX$.
Prove that $ \angle MTB - \angle CTM$ does not depend on choice of $ X$.
[i]Author: Farzan Barekat, Canada[/i]
Find a number $n \geq 9$ such that for any $n$ numbers, not necessarily distinct, $a_1,a_2, \ldots , a_n$, there exists 9 numbers $a_{i_1}, a_{i_2}, \ldots , a_{i_9}, (1 \leq i_1 < i_2 < \ldots < i_9 \leq n)$ and $b_i \in {4,7}, i =1, 2, \ldots , 9$ such that $b_1a_{i_1} + b_2a_{i_2} + \ldots + b_9a_{i_9}$ is a multiple of 9.