Let $f_1,f_2,\ldots,f_n : [0,1]\to (0,\infty)$ be $n$ continuous functions, $n\geq 1$, and let $\sigma$ be a permutation of the set $\{1,2,\ldots, n\}$. Prove that \[ \prod^n_{i=1} \int^1_0 \frac{ f_i^2(x) }{ f_{\sigma(i)}(x) } dx \geq \prod^n_{i=1} \int^1_0 f_i(x) dx. \]

In square $ABCD$, points $P$ and $Q$ lie on $\overline{AD}$ and $\overline{AB}$, respectively. Segments $\overline{BP}$ and $\overline{CQ}$ intersect at right angles at $R$, with $BR=6$ and $PR=7$. What is the area of the square? [asy] size(170); defaultpen(linewidth(0.6)); real r = 3.5; pair A = origin, B = (5,0), C = (5,5), D = (0,5), P = (0,r), Q = (5-r,0), R = intersectionpoint(B--P,C--Q); draw(A--B--C--D--A^^B--P^^C--Q^^rightanglemark(P,R,C,7)); dot("$A$",A,S); dot("$B$",B,S); dot("$C$",C,N); dot("$D$",D,N); dot("$Q$",Q,S); dot("$P$",P,W); dot("$R$",R,1.3*S); label("$7$",(P+R)/2,NE); label("$6$",(R+B)/2,NE); [/asy] $\textbf{(A) }85\qquad\textbf{(B) }93\qquad\textbf{(C) }100\qquad\textbf{(D) }117\qquad\textbf{(E) }125$

Let $G$ be a weighted bipartite graph $A \cup B$, with $|A| = |B| = n$. In other words, each edge in the graph is assigned a positive integer value, called its [i]weight.[/i] Also, define the weight of a perfect matching in $G$ to be the sum of the weights of the edges in the matching. Let $G'$ be the graph with vertex set $A \cup B$, and (which) contains the edge $e$ if and only if $e$ is part of some minimum weight perfect matching in $G$. Show that all perfect matchings in $G'$ have the same weight.

Prove that for each positive integer $K$ there exist infinitely many even positive integers which can be written in more than $K$ ways as the sum of two odd primes.

Let $D$ be an infinite in both sides sequence of $0$s and $1$s. For each positive integer $n$ we denote with $a_n$ the number of different subsequences of $0$s and $1$s in $D$ of length $n$. Does there exist a sequence $D$ for which for each $n\geq 22$ the number $a_n$ is equal to the $n$-th prime number?

Let $a,b,c$ be non-negative real numbers such that $a^2+b^2+c^2 \le 3$ then prove that; $$(a+b+c)(a+b+c-abc)\ge2(a^2b+b^2c+c^2a)$$

Suppose a triangle $\vartriangle ABC$ with $\angle C = 90^o$ is given. Let $D$ be the midpoint of $AC$, and let $E$ be the foot of the altitude through $C$ on $BD$. Show that the tangent in $C$ of the circumcircle of $\vartriangle AEC$ is perpendicular to $AB$.

Let $P$ and $Q$ be points taken inside of triangle $ABC$ such that $\angle APB=\angle AQC$ and $\angle APC=\angle AQB$. Circumcircle of $APQ$ intersects $AB$ and $AC$ second time at $K$ and $L$ respectively. Prove that $B,C,L,K$ are concyclic.

Two persons, $X$ and $Y$ , play with a die. $X$ wins a game if the outcome is $1$ or $2$; $Y$ wins in the other cases. A player wins a match if he wins two consecutive games. For each player determine the probability of winning a match within $5$ games. Determine the probabilities of winning in an unlimited number of games. If $X$ bets $1$, how much must $Y$ bet for the game to be fair ?

Jar A contains four liters of a solution that is $45\%$ acid. Jar B contains five liters of a solution that is $48\%$ acid. Jar C contains one liter of a solution that is $k\%$ acid. From jar C, $\tfrac{m}{n}$ liters of the solution is added to jar A, and the remainder of the solution in jar C is added to jar B. At the end, both jar A and jar B contain solutions that are $50\%$ acid. Given that $m$ and $n$ are relatively prime positive integers, find $k+m+n$.

Let $ S$ be a set with $ n$ elements, and $ F$ be a family of subsets of $ S$ with $ 2^{n\minus{}1}$ elements, such that for each $ A,B,C\in F$, $ A\cap B\cap C$ is not empty. Prove that the intersection of all of the elements of $ F$ is not empty.

Prove that the edges $AB$ and $CD$ of a tetrahedron $ABCD$ are perpendicular if and only if there exists a parallelogram $CDPQ$ such that $PA = PB = PD$ and $QA = QB = QC$.

Find the infinite integrals as follows. (1) 2013 Hiroshima City University entrance exam/Informatic Science $\int \frac{x^2}{2-x^2}dx$ (2) 2013 Kanseigakuin University entrance exam/Science and Technology $\int x^4\ln x\ dx$ (3) 2013 Shinsyu University entrance exam/Textile Science and Technology, Second-exam $\int \frac{\cos ^ 3 x}{\sin ^ 2 x}\ dx$

Determine all polynomials $P(x)$ with real coefficients such that \[(x+1)P(x-1)-(x-1)P(x)\] is a constant polynomial.

Two symbols $A$ and $B$ obey the rule $ABBB = B$. Given a word $x_1x_2\ldots x_{3n+1}$ consisting of $n$ letters $A$ and $2n+1$ letters $B$, show that there is a unique cyclic permutation of this word which reduces to $B$.

Define a sequence of integers by $T_1 = 2$ and for $n\ge2$, $T_n = 2^{T_{n-1}}$. Find the remainder when $T_1 + T_2 + \cdots + T_{256}$ is divided by 255. [i]Ray Li.[/i]

How many ways are there to make two $3$-digit numbers $m$ and $n$ such that $n=3m$ and each of six digits $1$, $2$, $3$, $6$, $7$, $8$ are used exactly once?

Evaluate \[\frac{1/2}{1+\sqrt2}+\frac{1/4}{1+\sqrt[4]2}+\frac{1/8}{1+\sqrt[8]2}+\frac{1/16}{1+\sqrt[16]2}+\cdots\] [i]Proposed by Ethan Tan[/i]

In a plane are given $n > 2$ distinct points. Some pairs of these points are connected by segments so that no two of the segments intersect. Prove that there are at most $3n-6$ segments.

Determine all triples of real numbers $(x, y, z)$ which satisfy the following system of equations: \[ \begin{cases} x+y+z=0 \\ x^3+y^3+z^3 = 90 \\ x^5+y^5+z^5=2850. \end{cases} \]

$A(-1,0),B(1,0)$. $D(x,0)$ is a point on $AB$. $CD\perp AB$, and $C$ is a point on unit circle. When $x\in$________, segments $AD,BD,CD$ can be three sides of a acute triangle.

Consider 2 concentric circle radii $ R$ and $ r$ ($ R > r$) with centre $ O.$ Fix $ P$ on the small circle and consider the variable chord $ PA$ of the small circle. Points $ B$ and $ C$ lie on the large circle; $ B,P,C$ are collinear and $ BC$ is perpendicular to $ AP.$ [b]i.)[/b] For which values of $ \angle OPA$ is the sum $ BC^2 \plus{} CA^2 \plus{} AB^2$ extremal? [b]ii.)[/b] What are the possible positions of the midpoints $ U$ of $ BA$ and $ V$ of $ AC$ as $ \angle OPA$ varies?

A set $S$ is constructed as follows. To begin, $S=\{0,10\}$. Repeatedly, as long as possible, if $x$ is an integer root of some polynomial $a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$ for some $n\geq 1$, all of whose coefficients $a_i$ are elements of $S$, then $x$ is put into $S$. When no more elements can be added to $S$, how many elements does $S$ have? $\textbf{(A) } 4 \qquad \textbf{(B) } 5 \qquad \textbf{(C) } 7 \qquad \textbf{(D) } 9 \qquad \textbf{(E) } 11$

Let $\vartriangle ABC$ be a triangle. Let $Q$ be a point in the interior of $\vartriangle ABC$, and let $X, Y,Z$ denote the feet of the altitudes from $Q$ to sides $BC$, $CA$, $AB$, respectively. Suppose that $BC = 15$, $\angle ABC = 60^o$, $BZ = 8$, $ZQ = 6$, and $\angle QCA = 30^o$. Let line $QX$ intersect the circumcircle of $\vartriangle XY Z$ at the point $W\ne X$. If the ratio $\frac{ WY}{WZ}$ can be expressed as $\frac{p}{q}$ for relatively prime positive integers $p, q$, find $p + q$.

Let \( \mathbb{N} \) be the set of all positive integers. We say that a function \( f: \mathbb{N} \to \mathbb{N} \) is Georgian if \( f(1) = 1 \) and, for every positive integer \( n \), there exists a positive integer \( k \) such that \[ f^{(k)}(n) = 1, \quad \text{where } f^{(k)} = f \circ f \cdots \circ f \quad \text{(applied } k \text{ times)}. \] If \( f \) is a Georgian function, we define, for each positive integer \( n \), \( \text{ord}(n) \) as the smallest positive integer \( m \) such that \( f^{(m)}(n) = 1 \). Determine all positive real numbers \( c \) for which there exists a Georgian function such that, for every positive integer \( n \geq 2024 \), it holds that \( \text{ord}(n) \geq cn - 1 \).