Found problems: 85335
A $3n$-table is a table with three rows and $n$ columns containing all the numbers $1, 2, …, 3n$. Such a table is called [i]tidy [/i] if the $n$ numbers in the first row appear in ascending order from left to right, and the three numbers in each column appear in ascending order from top to bottom. How many tidy $3n$-tables exist?
The value of
\[\frac{1}{20}-\frac{1}{21}+\frac{1}{20\times 21}\]
can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$.
[i]Proposed by Nathan Xiong[/i]
How many squares whose sides are parallel to the axes and whose vertices have coordinates that are integers lie entirely within the region bounded by the line $y=\pi x$, the line $y=-0.1$ and the line $x=5.1?$
$\textbf{(A)}\ 30 \qquad
\textbf{(B)}\ 41 \qquad
\textbf{(C)}\ 45 \qquad
\textbf{(D)}\ 50 \qquad
\textbf{(E)}\ 57$
Suppose positive real numers $x,y,z,w$ satisfy $(x^3+y^3)^4=z^3+w^3$. Prove that
$$x^4z+y^4w\geq zw.$$
Given a parallelogram $ABCD$ with $AB<BC$, show that the circumcircles of the triangles $APQ$ share a second common point (apart from $A$) as $P,Q$ move on the sides $BC,CD$ respectively s.t. $CP=CQ$.
Let $p,q$ be positive prime numbers and suppose $q>5$. Prove that if $q \mid 2^{p}+3^{p}$, then $q>p$.
[i]Laurentiu Panaitopol[/i]
Determine all functions $f : R \to R$ such that $f(x) \ge 0$ for all positive reals $x$, $f(0) = 0$ and for all reals $x, y$
$$f(x + y -xy) = f(x) + f(y) - f(xy).$$
Wen finds $17$ consecutive positive integers that sum to $2023$. Compute the smallest of these integers.
Find the range of the function $f(x)=\frac{\sqrt{x^2+1}}{x-1}$.
Solve the system of equations in integers:
$$\begin{cases} a^3=abc+2a+2c \\ b^3=abc-c \\ c^3=abc-a+b \end{cases}$$
Show that the equation $x^7 + y^7 = {1998}^z$ has no solution in positive integers.
Bisectrices $AA_1$ and $BB_1$ of triangle $ABC$ meet in $I$. Segments $A_1I$ and $B_1I$ are the bases of isosceles triangles with opposite vertices $A_2$ and $B_2$ lying on line $AB$. It is known that line $CI$ bisects segment $A_2B_2$. Is it true that triangle $ABC$ is isosceles?
Prove that if $ x,y$ are nonnegative real numbers with $ x\plus{}y\equal{}2$, then: $ x^2 y^2 (x^2\plus{}y^2) \le 2$.
Let $ABCD$ be a tetrahedron such that \[AB^2+CD^2=AC^2+BD^2=AD^2+BC^2.\] Show that at least one of its faces is an acute triangle.
Prove that the inequality $$\left(a_1+\cdots+a_n\right)\left(\frac{1}{a_1}+\cdots+\frac{1}{a_n}\right)\ge n^2$$ holds for any positive numbers $a_1,\ldots,a_n$ and determine when equality occurs.
Let $K$ be a point on the side $[AB]$, and $L$ be a point on the side $[BC]$ of the square $ABCD$. If $|AK|=3$, $|KB|=2$, and the distance of $K$ to the line $DL$ is $3$, what is $|BL|:|LC|$?
$ \textbf{(A)}\ \frac78
\qquad\textbf{(B)}\ \frac{\sqrt 3}2
\qquad\textbf{(C)}\ \frac 87
\qquad\textbf{(D)}\ \frac 38
\qquad\textbf{(E)}\ \frac{\sqrt 2}2
$
The excircle of a triangle $ABC$ corresponding to $A$ touches the side $BC$ at $K$ and the rays $AB$ and $AC$ at $P$ and $Q$, respectively. The lines $OB$ and $OC$ intersect $PQ$ at $M$ and $N$, respectively. Prove that
$$\frac{QN}{AB}=\frac{NM}{BC}=\frac{MP}{CA}.$$
A square with side $1$ is intersected by two parallel lines as shown in the figure. Find the sum of the perimeters of the shaded triangles if the distance between the lines is also $1$.
[img]https://cdn.artofproblemsolving.com/attachments/9/e/4e70610b80871325a72e923a0909eff06aebfa.png[/img]
Pablo, Sofia, and Mia got some candy eggs at a party. Pablo had three times as many eggs as Sofia, and Sofia had twice as many eggs as Mia. Pablo decides to give some of his eggs to Sofia and Mia so that all three will have the same number of eggs. What fraction of his eggs should Pablo give to Sofia?
$ \textbf{(A) }\dfrac{1}{12}\qquad\textbf{(B) }\dfrac{1}{6}\qquad\textbf{(C) }\dfrac{1}{4}\qquad\textbf{(D) }\dfrac{1}{3}\qquad\textbf{(E) }\dfrac{1}{2} $
At a party with $n\geq 4$ people, if every $3$ people have exactly $1$ common friend, how many different values can $n$ take?
$
\textbf{(A)}\ 1
\qquad\textbf{(B)}\ 2
\qquad\textbf{(C)}\ 4
\qquad\textbf{(D)}\ \text{Infinitely many}
\qquad\textbf{(E)}\ \text{None of the above}
$
Let $S$ be a set of $N \ge 3$ points in the plane. Assume that no $3$ points in $S$ are collinear. The segments with both endpoints in $S$ are colored in two colors.
Prove that there is a set of $N - 1$ segments of the same color which don't intersect except in their endpoints such that no subset of them forms a polygon with positive area.
Real numbers $a,b$ and $c$ satisfy the equalities $2015 (a + b + c) =1$ and $ab+bc+ca=2015 abc$.
Determine the numeric value of the expression $E=a^{2015}+b^{2015}+c^{2015}.$
$A$ is a $2m$-digit positive integer each of whose digits is $1$. $B$ is an $m$-digit positive integer each of whose digits is $4$. Prove that $A+B +1$ is a perfect square.
In their base $ 10$ representation, the integer $ a$ consists of a sequence of $ 1985$ eights and the integer $ b$ consists of a sequence of $ 1985$ fives. What is the sum of the digits of the base 10 representation of $ 9ab$?
$ \textbf{(A)}\ 15880 \qquad \textbf{(B)}\ 17856 \qquad \textbf{(C)}\ 17865 \qquad \textbf{(D)}\ 17874 \qquad \textbf{(E)}\ 19851$
Given a finite number of boys and girls, a [i]sociable set of boys[/i] is a set of boys such that every girl knows at least one boy in that set; and a [i]sociable set of girls[/i] is a set of girls such that every boy knows at least one girl in that set. Prove that the number of sociable sets of boys and the number of sociable sets of girls have the same parity. (Acquaintance is assumed to be mutual.)
[i](Poland) Marek Cygan[/i]