It is known that in a set of five coins three are genuine (and have the same weight) while two coins are fakes, each of which has a different weight from a genuine coin. What is the smallest number of weighings on a scale with two cups that is needed to locate one genuine coin?

How many natural numbers $(a,b,n)$ with $ gcd(a,b)=1$ and $ n>1 $ such that the equation \[ x^{an} +y^{bn} = 2^{2010} \] has natural numbers solution $ (x,y) $

Given $100$ coplanar points, no three collinear, prove that at most $70\%$ of the triangles formed by the points have all angles acute.

Complex numbers of apexes of 20-regular polygon inscribed to unit circle refer to are $Z_1,Z_2,\cdots,Z_{20}$ on complex plane. Then the number of points in $Z_1^{1995},Z_2^{1995},\cdots,Z_{20}^{1995}$ refer to is $\text{(A)}4\qquad\text{(B)}5\qquad\text{(C)}10\qquad\text{(D)}20$

Prove that if $\frac{p}{q}$ is an irreducible rational number that serves as a root of the polynomial $f(x) = a_0x^n + a_1x^{n-1} + ... + a_n$ with integer coefficients, then $p - kq$ is a divisor of $f(k)$ for any integer $k$.

Let $S$ be the set of all real polynomials $f(x) = ax^3 + bx^2 + cx + d$ such that $|f(x)| \le 1$ for all $ -1 \le x \le 1$. Show that the set of possible $|a|$ for $f$ in $S$ is bounded above and find the smallest upper bound.

In 3-dimensional space, two spheres centered at points $O_1$ and $O_2$ with radii $13$ and $20$ respectively intersect in a circle. Points $A, B, C$ lie on that circle, and lines $O_1A$ and $O_1B$ intersect sphere $O_2$ at points $D$ and $E$ respectively. Given that $O_1O_2 = AC = BC = 21,$ $DE$ can be expressed as $\frac{a\sqrt{b}}{c}$ where $a,b,c$ are positive integers. Find $a+b+c$.

Joshua rolls two dice and records the product of the numbers face up. The probability that this product is composite can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$. [i]Proposed by Nathan Xiong[/i]

An interior point in a rectangle is connected by line segments to the midpoints of its four sides. Thus four domains (polygons) with the areas $a, b, c$ and $d$ appear (see the figure). Prove that $a + c = b + d$. [img]https://1.bp.blogspot.com/-BipDNHELjJI/XzcCa68P3HI/AAAAAAAAMXY/H2Iqya9VItMLXrRqsdyxHLTXCAZ02nEtgCLcBGAsYHQ/s0/2002%2BMohr%2Bp1.png[/img]

Let $x, y, z$ be positive real numbers such that $x^2 + xy + y^2 = \frac{25}{4}$, $y^2 + yz + z^2 = 36$, and $z^2 + zx + x^2 = \frac{169}{4}$. Find the value of $xy + yz + zx$.

Given cube $ ABCD.EFGH $ with $ AB = 4 $ and $ P $ midpoint of the side $ EFGH $. If $ M $ is the midpoint of $ PH $, find the length of segment $ AM $.

Positive intenger $n\geq3$. $a_1,a_2,\cdots,a_n$ are $n$ positive intengers that are pairwise coprime, satisfying that there exists $k_1,k_2,\cdots,k_n\in\{-1,1\}, \sum_{i=1}^{n}k_ia_i=0$. Are there positive intengers $b_1,b_2,\cdots,b_n$, for any $k\in\mathbb{Z}_+$, $b_1+ka_1,b_2+ka_2,\cdots,b_n+ka_n$ are pairwise coprime?

We are given a right triangle $ABC$ and the median $BD$ drawn from the vertex $B$ of the right angle. Let the circle inscribed in $\vartriangle ABD$ be tangent to side $AD$ at $K$. Find the angles of $\vartriangle ABC$ if $K$ divides $AD$ in halves.

Let $ f:\mathbb{R}\longrightarrow\mathbb{R} ,f(x)=\left\{\begin{matrix} \sin x , & x\not\in\mathbb{Q} \\ 0, & x\in\mathbb{Q}\end{matrix}\right. . $ [b]a)[/b] Determine the maximum length of an interval $ I\subset\mathbb{R} $ such that $ f|_I $ is discontinuous everywhere, yet has the intermediate value property. [b]b)[/b] Study the convergence of the sequence $ \left( x_n\right)_{n\in\mathbb{N}\cup\{ 0\}} $ defined by $ x_0\in (0,\pi /2),x_{n+1}=f\left( x_n\right),\forall n\ge 0. $

How many whole numbers lie in the interval between $\frac{5}{3}$ and $2\pi$? $\textbf{(A)}\ 2\qquad \textbf{(B)}\ 3\qquad \textbf{(C)}\ 4\qquad \textbf{(D)}\ 5\qquad \textbf{(E)}\ \text{infinitely many}$

Find, with proof, all nonconstant polynomials $P(x)$ with real coefficients such that, for all nonzero real numbers $z$ with $P(z)\neq 0$ and $P(\frac{1}{z}) \neq 0$ we have $$\frac{1}{P(z)}+\frac{1}{P(\frac{1} {z})}=z+\frac{1}{z}.$$

Show that the equation $2x^2-3x=3y^2$ has infinitely many solutions in positive integers.

For a finite nonempty set $A$ of positive integers, $A =\{a_1, a_2,\dots , a_n\}$, we say the calamitous complement of A is the set of all positive integers $k$ for which there do not exist nonnegative integers $w_1, w_2, \dots , w_n$ with $k = a_1w_1 + a_2w_2 +\dots + a_nw_n.$ The calamitous complement of $A$ is denoted $cc(A)$. For example, $cc(\{5, 6, 9\}) = \{1, 2, 3, 4, 7, 8, 13\}$. Find all pairs of positive integers $a, b$ with $1 < a < b$ for which there exists a set $G$ satisfying all of the following properties: 1. $G$ is a set of at most three positive integers, 2. $cc(\{a, b\})$ and $cc(G)$ are both finite sets, and 3. $cc(G) = cc(\{a, b\})\cup \{m\}$ for some $m$ not in $cc(\{a, b\})$.

A rectangle has side lengths $6$ and $8$. There are relatively prime positive integers $m$ and $n$ so that $\tfrac{m}{n}$ is the probability that a point randomly selected from the inside of the rectangle is closer to a side of the rectangle than to either diagonal of the rectangle. Find $m + n$.

Let $m$ and $n$ be positive integers such that $5$ divides $2^n + 3^m$. Prove that $5$ divides $2^m + 3^n$.

How many different permutations $(\alpha_1 \alpha_2\alpha_3\alpha_4\alpha_5)$ of the set $\{1,2,3,4,5\}$ are there such that $(\alpha_1\dots \alpha_k)$ is not a permutation of the set $\{1,\dots ,k\}$, for every $1\leq k \leq 4$? $ \textbf{(A)}\ 13 \qquad\textbf{(B)}\ 65 \qquad\textbf{(C)}\ 71 \qquad\textbf{(D)}\ 461 \qquad\textbf{(E)}\ \text{None} $

There are $13$ stones each of which weighs an integer number of grams. It is known that any $12$ of them can be put on two pans of a balance scale, six on each pan, so that they are in equilibrium (i.e., each pan will carry an equal total weight). Prove that all stones weigh the same number of grams.

Let $f$ be a non-constant function from the set of positive integers into the set of positive integer, such that $a-b$ divides $f(a)-f(b)$ for all distinct positive integers $a$, $b$. Prove that there exist infinitely many primes $p$ such that $p$ divides $f(c)$ for some positive integer $c$. [i]Proposed by Juhan Aru, Estonia[/i]

Rational numbers $a$, $b$, $c$ satisfy the equation \[(a+b+c)(a+b-c)=c^2\,.\] Show that $a+b=c=0$.

Find the maximum positive integer $k$ such that for any positive integers $m,n$ such that $m^3+n^3>(m+n)^2$, we have $$m^3+n^3\geq (m+n)^2+k$$ [i] Proposed by Dorlir Ahmeti, Albania[/i]