How many integers $ N$ less than 1000 can be written as the sum of $ j$ consecutive positive odd integers from exactly 5 values of $ j\ge 1$?

A college student is about to break up with her boyfriend, a mathematics major who is apparently more interested in math than her. Frustrated, she cries, ”You mathematicians have no soul! It’s all numbers and equations! What is the root of your incompetence?!” Her boyfriend assumes she means the square root of himself, or the square root of i. What two answers should he give?

Find all functions $f: R\to R$ such that: \[ f(x^2)-f(y^2)=(x+y)(f(x)-f(y)), x,y \in R \]

How many kites are there such that all of its four vertices are vertices of a given regular icosagon ($20$-gon)? $ \textbf{(A)}\ 105 \qquad\textbf{(B)}\ 100 \qquad\textbf{(C)}\ 95 \qquad\textbf{(D)}\ 90 \qquad\textbf{(E)}\ 85 $

Let $a, b, c$ be positive real numbers such that $a + b + c = 1$. Prove that $$2 \left( \frac{ab}{a + b} +\frac{bc}{b + c} +\frac{ca}{c+ a}\right)+ 1 \ge 6(ab + bc + ca)$$ Trần Nam Dũng

Show that a convex polyhedron with an odd number of faces has at least one face with an even number of edges.

Solve the equation \[(x+1)\log^2_{3}x+4x\log_{3}x-16=0\]

Some time ago, on a radio program, a baker announced a special promotion in the purchase of two stuffed cakes. Each cake could contain up to five fillings of which had in the pastry. On the show, a lady said there were $1,048,576$ different possibilities to choose the two stuffed cakes. How many different fillings did the pastry chef have?

Let $S$ be a point outside of the plane of the parallelogram $ABCD$, such that the triangles $SAB$, $SBC$, $SCD$ and $SAD$ are equivalent. a) Prove that $ABCD$ is a rhombus. b) If the distance from $S$ to the plane $(A, B, C, D)$ is $12$, $BD = 30$ and $AC = 40$, compute the distance from the projection of the point $S$ on the plane $(A, B, C, D)$ to the plane $(S,B,C)$ .

Diameter $AB$ is drawn to a circle with radius $1$. Two straight lines $s$ and $t$ touch the circle at points $A$ and $B$, respectively. Points $P$ and $Q$ are chosen on the lines $s$ and $t$, respectively, so that the line $PQ$ touches the circle. Find the smallest possible area of the quadrangle $APQB$.

Given a set of boys and girls, we call a pair $(A,B)$ amicable if $A$ and $B$ are friends. The friendship relation is symmetric. A set of people is affectionate if it satisfy the following conditions: i) The set has the same number of boys and girls. ii) For every four different people $A,B,C,D$ if the pairs $(A,B),(B,C),(C,D)$ and $(D,A)$ are all amicable, then at least one of the pairs $(A,C)$ and $(B,D)$ is also amicable. iii) At least $\frac{1}{2013}$ of all boy-girl pairs are amicable. Let $m$ be a positive integer. Prove that there exists an integer $N(m)$ such that if a affectionate set has al least $N(m)$ people, then there exists $m$ boys that are pairwise friends or $m$ girls that are pairwise friends.

Determine all positive integers $n$ such that all positive integers less than $n$ and coprime to $n$ are powers of primes.

Prove that there are infinitely many primes representable in the form $m^2+mn+n^2$ for some integers $m,n$ .

Find all natural numbers $ n$,such that there exist $ x_1,x_2,\dots,x_{n\plus{}1}\in\mathbb{N}$,such that $ \frac{1}{x_1^2}\plus{}\frac{1}{x_2^2}\plus{}\dots\plus{}\frac{1}{x_n^2}\equal{}\frac{n\plus{}1}{x_{n\plus{}1}^2}$.

Given a cyclic quadrilateral $ABCD$, define $E$ as $AD \cap BC$ and $F$ as $AB \cap CD$. Let $\Omega_A$ be the circle passing through $A, D$ and tangent to $AB$, and let its center be $O_A$. Let $\Gamma_B$ be the circle passing through $B, C$ and tangent to $AB$, and let its center be $O_B$. Let $\Gamma_C$ be the circle passing through $B, C$ and tangent to $CD$, and let its center be $O_C$. Let $\Omega_D$ be the circle passing through $A, D$ and tangent to $CD$, and let its center be $O_D$. Prove that $O_AO_BO_CO_D$ is cyclic, and prove that it's center lies on $EF$.

Define the polynomials $P_0, P_1, P_2 \cdots$ by: \[ P_0(x)=x^3+213x^2-67x-2000 \] \[ P_n(x)=P_{n-1}(x-n), n \in N \] Find the coefficient of $x$ in $P_{21}(x)$.

Is it possible to write a positive integer in every cell of an infinite chessboard, in such a manner that, for all positive integers $m, n$, the sum of numbers in every $m\times n$ rectangle is divisible by $m + n$ ?

Let $a, b, c, d > 0$ satisfying $abcd = 1$. Prove that $$\frac{1}{a + b + 2}+\frac{1}{b + c + 2}+\frac{1}{c + d + 2}+\frac{1}{d + a + 2} \le 1$$

Find all triples $(p, q, r)$ of prime numbers for which $4q - 1$ is a prime number and $$\frac{p + q}{p + r} = r - p$$ holds. [i](Walther Janous)[/i]

Jingyuan is designing a bucket hat for BMT merchandise. The hat has the shape of a cylinder on top of a truncated cone, as shown in the diagram below. The cylinder has radius $9$ and height $12$. The truncated cone has base radius $15$ and height $4$, and its top radius is the same as the cylinder’s radius. Compute the total volume of this bucket hat. [img]https://cdn.artofproblemsolving.com/attachments/a/9/467d19889d08a6081f9dcd3f4d9df60582f244.png[/img]

Let $P$ be a point on a circle $C$ and let $\phi$ be a given angle incommensurable with $2\pi$. For each $n \in N, P_n$ denotes the image of $P$ under the rotation about the center $O$ of $C$ by the angle $\alpha_n = n \phi$. Prove that the set $M = \{P_n | n \ge 0\}$ is dense in $C$.

An infinite triangular lattice is given, such that the distance between any two adjacent points is always equal to $1$. Points $A$, $B$, and $C$ are chosen on the lattice such that they are the vertices of an equilateral triangle of side length $L$, and the sides of $ABC$ contain no points from the lattice. Prove that, inside triangle $ABC$, there are exactly $\frac{L^2-1}{2}$ points from the lattice.

Let $ x, y, z$ be positive real numbers such that $ x^n, y^n$ and $ z^n$ are side lengths of some triangle for all positive integers $ n$. Prove that at least two of x, y and z are equal.

Let $ABC$ be a triangle with $\angle C = 90^o$ and $AC = 1$. The median $AM$ intersects the incircle at the points $P$ and $Q$, with $P$ between $A$ and $Q$, such that $AP = QM$. Find the length of $PQ$.

At least $ n - 1$ numbers are removed from the set $\{1, 2, \ldots, 2n - 1\}$ according to the following rules: (i) If $ a$ is removed, so is $ 2a$; (ii) If $ a$ and $ b$ are removed, so is $ a \plus{} b$. Find the way of removing numbers such that the sum of the remaining numbers is maximum possible.