Found problems: 85335
Let $n$ be a positive integer. Find the maximal number of non-congruent triangles whose sides lengths are integers $\leq n.$
Let $P(x),\ Q(x)$ be polynomials such that :
\[\int_0^2 \{P(x)\}^2dx=14,\ \int_0^2 P(x)dx=4,\ \int_0^2 \{Q(x)\}^2dx=26,\ \int_0^2 Q(x)dx=2.\]
Find the maximum and the minimum value of $\int_0^2 P(x)Q(x)dx$.
There are totally $16$ teams participating in a football tournament, each team playing with every other exactly $1$ time. In each match, the winner gains $3$ points, the loser gains $0$ point and each teams gain $1$ point for the tie match. Suppose that at the end of the tournament, each team gains the same number of points. Prove that there are at least $4$ teams that have the same number of winning matches, the same number of losing matches and the same number of tie matches.
At the base of the quadrangular pyramid $SABCD$ lies the quadrangle $ABCD$. whose diagonals are perpendicular and intersect at point $P$, and $SP$ is the altitude of the pyramid. Prove that the projections of the point $P$ onto the lateral faces of the pyramid lie on the same circle.
(A. Zaslavsky)
A gambling student tosses a fair coin. She gains $1$ point for each head that turns up, and gains $2$ points for each tail that turns up. Prove that the probability of the student scoring [i]exactly[/i] $n$ points is $\frac{1}{3}\cdot\left(2+\left(-\frac{1}{2}\right)^{n}\right)$.
Let $I$ be the incenter of a triangle $ABC$. The points $X, Y$ lie on the prolongations of the lines $IB, IC$ after $I$ so that $\angle IAX=\angle IBA$ and $\angle IAY=\angle ICA$. Show that the line through the midpoints of $IA$ and $XY$ passes through the circumcenter of $ABC$.
In a triangle $ABC$, let $I$ denote its incenter. Points $D, E, F$ are chosen on the segments $BC, CA, AB$, respectively, such that $BD + BF = AC$ and $CD + CE = AB$. The circumcircles of triangles $AEF, BFD, CDE$ intersect lines $AI, BI, CI$, respectively, at points $K, L, M$ (different from $A, B, C$), respectively. Prove that $K, L, M, I$ are concyclic.
Determine the prime numbers $p$ and $q$ that satisfy the equality: $p^3 + 107 = 2q (17q + 24)$ .
In the base ten number system the number $526$ means $5 \cdot 10^2+2 \cdot 10 + 6$. In the Land of Mathesis, however, numbers are written in the base $r$. Jones purchases an automobile there for $440$ monetary units (abbreviated m.u). He gives the salesman a $1000$ m.u bill, and receives, in change, $340$ m.u. The base $r$ is:
${{ \textbf{(A)}\ 2\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 7\qquad\textbf{(D)}\ 8}\qquad\textbf{(E)}\ 12} $
In triangle $ ABC$, $ AB \equal{} AC \equal{} 100$, and $ BC \equal{} 56$. Circle $ P$ has radius $ 16$ and is tangent to $ \overline{AC}$ and $ \overline{BC}$. Circle $ Q$ is externally tangent to $ P$ and is tangent to $ \overline{AB}$ and $ \overline{BC}$. No point of circle $ Q$ lies outside of $ \triangle ABC$. The radius of circle $ Q$ can be expressed in the form $ m \minus{} n\sqrt {k}$, where $ m$, $ n$, and $ k$ are positive integers and $ k$ is the product of distinct primes. Find $ m \plus{} nk$.
Compute the largest possible radius of a circle contained in the region defined by $|x+|y|| \le 1$ in the coordinate plane.
Find the number of $f:\{1,\ldots, 5\}\to \{1,\ldots, 5\}$ such that $f(f(x))=x$
(A)$26$
(B)$41$
(C)$120$
(D)$60$
Triangle $ABC$ has a right angle at $C$. Let $D$ be the midpoint of side $\overline{AC}$, and let $E$ be the intersection of $\overline{AC}$ and the bisector of $\angle ABC$. The area of $\triangle ABC$ is $144$, and the area of $\triangle DBE$ is $8$. Find $AB^2$.
Given that $r$ and $s$ are relatively prime positive integers such that $\dfrac{r}{s}=\dfrac{2(\sqrt{2}+\sqrt{10})}{5\left(\sqrt{3+\sqrt{5}}\right)}$, find $r$ and $s$.
Two of the diagonals of a regular pentagon are selected at random. What is the probability that the two selected diagonals intersect inside the pentagon?
$\text{(A) }\frac{2}{5}\qquad\text{(B) }\frac{1}{5}\qquad\text{(C) }\frac{7}{10}\qquad\text{(D) }\frac{3}{5}\qquad\text{(E) }\frac{1}{2}$
Consider the integers formed using the digits $0,1,2,3,4,5,6$, without repetition. Find the largest multiple of $55$. Justify your answer.
Find the number of sets $A$ that satisfy the three conditions:
$\star$ $A$ is a set of two positive integers
$\star$ each of the numbers in $A$ is at least $22$ percent the size of the other number
$\star$ $A$ contains the number $30.$
Prove that for $\forall$ $a,b,c\in [\frac{1}{3},3]$ the following inequality is true:
$\frac{a}{a+b}+\frac{b}{b+c}+\frac{c}{c+a}\geq \frac{7}{5}$.
The enemy ship has landed on a $9\times 9$ board that covers exactly $5$ squares of the board, like this:
[img]https://cdn.artofproblemsolving.com/attachments/2/4/ae5aa95f5bb5e113fd5e25931a2bf8eb872dbe.png[/img]
The ship is invisible. Each defensive missile covers exactly one square, and destroys the ship if it hits one of the $5$ squares that it occupies. Determine the minimum number of missiles needed to destroy the enemy ship with certainty .
A BWM tree is defined recursively:
[list]
[*] An empty tree is a BWM tree of height 0 and size 0.
[*] A nonempty BWM tree consists of a root node and three subtrees, each of which is itself a (possibly empty) BWM tree. The height of the tallest of the subtrees must be at most 2 more than the height of the shortest.
[*] The height of a nonempty BWM tree is one more than the height of its tallest subtree, and the size of a nonempty BWM tree is one more than the sum of the sizes of the subtrees.
[/list]
What is the minimum size of a height-10 BWM tree?
[i]Proposed by Jacob Weiner[/i]
Can you form six squares with nine matches? How about fourteen squares with eight matches? (It is assumed that all matches have equal length, and you cannot break any of them.)
Let $n>1$ be an integer. Prove that there exists an integer $n-1 \ge m \ge \left \lfloor \frac{n}{2} \right \rfloor$ such that the following equation has integer solutions with $a_m>0:$
$$\frac{a_{m}}{m+1}+\frac{a_{m+1}}{m+2}+ \cdots + \frac{a_{n-1}}{n}=\frac{1}{\textrm{lcm}\left ( 1,2, \cdots , n \right )}$$
[i]Proposed by Navid Safaei[/i]
We have a convex quadrilateral $ABCD$ and a point $M$ on its side $AD$ such that $CM$ and $BM$ are parallel to $AB$ and $CD$ respectively. Prove that $S_{ABCD} \geq 3 S_{BCM}.$
[i]Remark.[/i] $S$ denotes the area function.
We say that a subset $A$ of $\mathbb N$ is good if for some positive integer $n$, the equation $x-y=n$ admits infinitely many solutions with $x,y\in A$. If $A_1,A_2,\ldots,A_{100}$ are sets whose union is $\mathbb N$, prove that at least one of the $A_i$s is good.
A matrix $A=(a_{ij})$ is called [i]nice[/i], if it has the following properties:
(i) the set of all entries of $A$ is $\{1,2,\dots,2t\}$ for some integer $t$;
(ii) the entries are non-decreasing in every row and in every column: $a_{i,j} \le a_{i,j+1}$ and $a_{i,j} \le a_{i+1,j}$;
(iii) equal entries can appear only in the same row or the same column: if $a_{i,j}=a_{k,\ell}$, then either $i=k$ or $j=\ell$;
(iv) for each $s=1,2,\dots,2t-1$, there exist $i \ne k$ and $j \ne \ell$ such that $a_{i,j}=s$ and $a_{k,\ell}=s+1$.
Prove that for any positive integers $m$ and $n$, the number of nice $m \times n$ matrixes is even.
For example, the only two nice $2 \times 3$ matrices are $\begin{pmatrix} 1 & 1 & 1\\2 & 2 & 2 \end{pmatrix}$ and $\begin{pmatrix} 1 & 1 & 3\\2 & 4 & 4 \end{pmatrix}$.