How many ordered four-tuples of integers $(a,b,c,d)$ with $0 < a < b < c < d < 500$ satisfy $a + d = b + c$ and $bc - ad = 93$?

An altitude $h$ of a triangle is increased by a length $m$. How much must be taken from the corresponding base $b$ so that the area of the new triangle is one-half that of the original triangle? ${{ \textbf{(A)}\ \frac{bm}{h+m}\qquad\textbf{(B)}\ \frac{bh}{2h+2m}\qquad\textbf{(C)}\ \frac{b(2m+h)}{m+h}\qquad\textbf{(D)}\ \frac{b(m+h)}{2m+h} }\qquad\textbf{(E)}\ \frac{b(2m+h)}{2(h+m)} } $

Suppose $n$ is an integer $\geq 2$. Determine the first digit after the decimal point in the decimal expansion of the number \[\sqrt[3]{n^{3}+2n^{2}+n}\]

Let $\vartriangle ABC$ be a triangle of which the side lengths are positive integers which are pairwise coprime. The tangent in $A$ to the circumcircle intersects line $BC$ in $D$. Prove that $BD$ is not an integer.

Prove that every natural number which is not an integer power of $2$ is the sum of two or more consecutive natural numbers.

Two disjoint circles $W_1(S_1,r_1)$ and $W_2(S_2,r_2)$ are given in the plane. Point $A$ is on circle $W_1$ and $AB,AC$ touch the circle $W_2$ at $B,C$ respectively. Find the loci of the incenter and orthocenter of triangle $ABC$.

Positive numbers $a, b, c, x, y, z$ satisfy that $cy + bz = a$, $az + cx = b$, and $bx + ay = c$. Find the minimum value of the function $f(x,y,z) =\frac{x^2}{x+1}+\frac{y^2}{y+1}+\frac{z^2}{z+1}$.

$m,n,p$ are positive integers which do not simultaneously equal to zero. $3$D Cartesian space is divided into unit cubes by planes each perpendicular to one of $3$ axes and cutting corresponding axis at integer coordinates. Each unit cube is filled with an integer from $1$ to $60$. A filling of integers is called [i]Dien Bien[/i] if, for each rectangular box of size $\{2m+1,2n+1,2p+1\}$, the number in the unit cube which has common centre with the rectangular box is the average of the $8$ numbers of the $8$ unit cubes at the $8$ corners of that rectangular box. How many [i]Dien Bien[/i] fillings are there? Two fillings are the same if one filling can be transformed to the other filling via a translation. [hide]translation from [url=http://artofproblemsolving.com/community/c6h592875p3515526]here[/url][/hide]

Math teacher wrote in a table a polynomial $P(x)$ with integer coefficients and he said: "Today my daughter have a birthday.If in polynomial $P(x)$ we have $x=a$ where $a$ is the age of my daughter we have $P(a)=a$ and $P(0)=p$ where $p$ is a prime number such that $p>a$." How old is the daughter of math teacher?

Find all triplets of real numbers $(x, y, z)$ that satisfies the system of equations $x^5 = 2y^3 + y - 2$ $y^5 = 2z^3 + z - 2$ $z^5 = 2x^3 + x - 2$

consider $0<u<1$. find $\alpha > 0$ minimum such that there exists $\beta > 0$ satisfying $(1+x)^u +(1-x)^u \leq 2 - \frac{x^\alpha}{\beta} \forall 0<x<1$

Find all integer coefficient polynomial $P(x)$ such that for all positive integer $x$, we have $$\tau(P(x))\geq\tau(x)$$Where $\tau(n)$ denotes the number of divisors of $n$. Define $\tau(0)=\infty$. Note: you can use this conclusion. For all $\epsilon\geq0$, there exists a positive constant $C_\epsilon$ such that for all positive integer $n$, the $n$th smallest prime is at most $C_\epsilon n^{1+\epsilon}$. [i]Proposed by USJL[/i]

Let $a$ and $ b$ be acute corners. Prove that if $\sin a$, $\sin b$, and $\sin (a + b)$ are rational numbers, then $\cos a$, $\cos b$, and $\cos (a + b)$ are also rational numbers.

Let $\lfloor x \rfloor$ denote the greatest integer less than or equal to $x$. Find the number of positive integers $m$ between $1$ and $2022$ inclusive such that \[ \left\lfloor \frac{3^m}{11} \right\rfloor \] is even.

Suppose that $(\underline{AB}, \underline{CD})$ is a pair of two digit positive integers (digits $A$ and $C$ must be nonzero) such that the product $\underline{AB} \cdot \underline{CD}$ divides the four digit number $\underline{ABCD}$. Find the sum of all possible values of the three digit number $\underline{ABC}$.

Consider $\triangle MAB$ with a right angle at $A$ and circumcircle $\omega$. Take any chord $CD$ perpendicular to $AB$ such that $A, C, B, D, M$ lie on $\omega$ in this order. Let $AC$ and $MD$ intersect at point $E$, and let $O$ be the circumcenter of $\triangle EMC$. Show that if $J$ is the intersection of $BC$ and $OM$, then $JB = JM$. [i](Proposed by Matthew Kung Wei Sheng and Ivan Chan Kai Chin)[/i]

Let $M$ be the intersection point of the diagonals of the parallelogram $ABCD$. Consider three circles passing through $M$, the first and second touch $AB$ at points $A$ and $B$, respectively, and the third passes through $C$ and $D$. Let us denote by $P$ and $C$, respectively, the intersection points of the first circle with the third and the second with the third, different from $M$. Prove that the line $PQ$ touches the first and second circles.

Determine whether there exists a differentiable function $f:[0,1]\to\mathbb R$ such that $$f(0)=f(1)=1,\qquad|f'(x)|\le2\text{ for all }x\in[0,1]\qquad\text{and}\qquad\left|\int^1_0f(x)dx\right|\le\frac12.$$

It is possible to paint with the colors red and blue the squares of a grid board $1999\times 1999$, so that in each of the $1999$ rows, in each of the $1999$ columns and each of the the $2$ diagonals are exactly $1000$ squares painted red?

In the polygon shown, each side is perpendicular to its adjacent sides, and all 28 of the sides are congruent. The perimeter of the polygon is $56$. The area of the region bounded by the polygon is [asy] draw((0,0)--(1,0)--(1,-1)--(2,-1)--(2,-2)--(3,-2)--(3,-3)--(4,-3)--(4,-2)--(5,-2)--(5,-1)--(6,-1)--(6,0)--(7,0)--(7,1)--(6,1)--(6,2)--(5,2)--(5,3)--(4,3)--(4,4)--(3,4)--(3,3)--(2,3)--(2,2)--(1,2)--(1,1)--(0,1)--cycle); [/asy] $ \textbf{(A)}\ 84 \qquad\textbf{(B)}\ 96 \qquad\textbf{(C)}\ 100 \qquad\textbf{(D)}\ 112 \qquad\textbf{(E)}\ 196 $

Under the assumption that the following set of relations has a unique solution for $u(t),$ determine it. $$ \frac{d u(t) }{dt} = u(t) + \int_{0}^{t} u(s)\, ds, \;\;\; u(0)=1.$$

Determine the largest integer $n$ such that $n < 103$ and $n^3 - 1$ is divisible by $103$.

Let $I$ be the incenter of a triangle $ABC$. The circle passing through $I$ and centered at $A$ meets the circumference of the triangle $ABC$ at points $M$ and $N$. Prove that the line $MN$ touches the incircle of the triangle $ABC$. (I. Kachan)

Associate to any point $ (h,k) $ in the integer net of the cartesian plane a real number $ a_{h,k} $ so that $$ a_{h,k}=\frac{1}{4}\left( a_{h-1,k} +a_{h+1,k}+a_{h,k-1}+a_{h,k+1}\right) ,\quad\forall h,k\in\mathbb{Z} . $$ [b]a)[/b] Prove that it´s possible that all the elements of the set $ A:=\left\{ a_{h,k}\big| h,k\in\mathbb{Z}\right\} $ are different. [b]b)[/b] If so, show that the set $ A $ hasn´t any kind of boundary.

Let $\triangle P_1P_2P_3$ be an equilateral triangle. For each $n\ge 4$, [i]Mingmingsan[/i] can set $P_n$ as the circumcenter or orthocenter of $\triangle P_{n-3}P_{n-2}P_{n-1}$. Find all positive integer $n$ such that [i]Mingmingsan[/i] has a strategy to make $P_n$ equals to the circumcenter of $\triangle P_1P_2P_3$. [i]Proposed by Li4 and Untro368.[/i]