What percentage of twenty thousand is a quarter of a million?

Let $ABC$ be an acute angled triangle. Let $B' , A'$ be points on the perpendicular bisectors of $AC, BC$ respectively such that $B'A \perp AB$ and $A'B \perp AB$. Let $P$ be a point on the segment $AB$ and $O$ the circumcenter of the triangle $ABC$. Let $D, E$ be points on $BC, AC$ respectively such that $DP \perp BO$ and $EP \perp AO$. Let $O'$ be the circumcenter of the triangle $CDE$. Prove that $B', A'$ and $O'$ are collinear. [i]Steve Dinh[/i]

Let $AD, BE, CF$ be segments whose midpoints are on the same line $\ell$. The points $X, Y, Z$ lie on the lines $EF, FD, DE$ respectively such that $AX \parallel BY \parallel CZ \parallel \ell$. Prove that $X, Y, Z$ are collinear. Tran Quang Hung, High School of Natural Sciences, Hanoi National University

A girls' camp is located $ 300$ rods from a straight road. On this road, a boys' camp is located $ 500$ rods from the girls' camp. It is desired to build a canteen on the road which shall be exactly the same distance from each camp. The distance of the canteen from each of the camps is: $ \textbf{(A)}\ 400\text{ rods} \qquad\textbf{(B)}\ 250\text{ rods} \qquad\textbf{(C)}\ 87.5\text{ rods} \qquad\textbf{(D)}\ 200\text{ rods}\\ \textbf{(E)}\ \text{none of these}$

Let $ABC$ be an equilateral triangle of side length $1.$ For a real number $0<x<0.5,$ let $A_1$ and $A_2$ be the points on side $BC$ such that $A_1B=A_2C=x,$ and let $T_A=\triangle AA_1A_2.$ Construct triangles $T_B=\triangle BB_1B_2$ and $T_C=\triangle CC_1C_2$ similarly. There exist positive rational numbers $b,c$ such that the region of points inside all three triangles $T_A,T_B,T_C$ is a hexagon with area $$\dfrac{8x^2-bx+c}{(2-x)(x+1)}\cdot \dfrac{\sqrt 3}{4}.$$ Find $(b,c).$

Let $\frac{2}{\sqrt5+1}\leq p < 1$, and let the real sequence $\{ a_n \}$ have the following property: for every sequence $\{ e_n \}$ of $0$'s and $\pm 1$'s for which $\sum_{n=1}^\infty e_np^n=0$, we also have $\sum_{n=1}^\infty e_na_n=0$. Prove that there is a number $c$ such that $a_n=cp^n$ for all $n$. [Z. Daroczy, I. Katai]

For each $t \geq 0$ let $S_t$ be the set of all nonnegative, increasing, convex, continuous, real-valued functions $f(x)$ defined on the closed interval $[0,1]$ for which $$f(1) -2 f(2 \slash 3) +f (1 \slash 3) \geq t( f( 2 \slash 3) -2 f(1 \slash 3) +f(0)).$$ Define necessary and sufficient conditions on $ t$ for $S_t $ to be closed under multiplication.

Let $\mathbb{R}$ be the set of real numbers. Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a function such that \[f(x+y)f(x-y)\geqslant f(x)^2-f(y)^2\] for every $x,y\in\mathbb{R}$. Assume that the inequality is strict for some $x_0,y_0\in\mathbb{R}$. Prove that either $f(x)\geqslant 0$ for every $x\in\mathbb{R}$ or $f(x)\leqslant 0$ for every $x\in\mathbb{R}$.

In a soccer league with $2020$ teams every two team have played exactly once and no game have lead to a draw. The participating teams are ordered first by their points (3 points for a win, 1 point for a draw, 0 points for a loss) then by their goal difference (goals scored minus goals against) in a normal soccer table. Is it possible for the goal difference in such table to be strictly increasing from the top to the bottom? [i]Proposed by Abolfazl Asadi[/i]

Points $E$ and $F$ lie on the sides $AB$ and $AC$ of a triangle $ABC$. Lines $EF$ and $BC$ meet at point $S$. Let $M$ and $N$ be the midpoints of $BC$ and $EF$, respectively. The line passing through $A$ and parallel to $MN$ meets $BC$ at point $K$. Prove that $\frac{BK}{CK}=\frac{FS}{ES}$ . .

First $a$ is chosen at random from the set $\{1,2,3,\ldots,99,100 \}$, and then $b$ is chosen at random from the same set. The probability that the integer $3^a+7^b$ has units digit $8$ is $\text{(A)} \ \frac1{16} \qquad \text{(B)} \ \frac18 \qquad \text{(C)} \ \frac{3}{16}\qquad \text{(D)} \ \frac15 \qquad \text{(E)} \ \frac14$

Zeros of a fourth-degree polynomial $f (x)$ form an arithmetic progression. Prove that the zeros of $f '(x)$ also form an arithmetic progression.

Let $a_1,a_2,...,a_{2007}$ be real number such that $ a_1+a_2+...+a_{2007}\geq 2007^{2}$ and $a_1^{2}+a_2^{2}+...+a_{2007}^{2}\leq 2007^{3}-1 $. Prove that $ a_k\in[2006;2008]$ for all $k\in\left \{ 1,2,...,2007 \right \}$

A quadrilateral $ABCD$ is inscribed into a circle $\omega$ with center $O$. Let $M_1$ and $M_2$ be the midpoints of segments $AB$ and $CD$ respectively. Let $\Omega$ be the circumcircle of triangle $OM_1M_2$. Let $X_1$ and $X_2$ be the common points of $\omega$ and $\Omega$ and $Y_1$ and $Y_2$ the second common points of $\Omega$ with the circumcircles of triangles $CDM_1$ and $ABM_2$. Prove that $X_1X_2 // Y_1Y_2$.

Find the complex numbers $x,y,z$,with $\mid x\mid=\mid y\mid=\mid z\mid$,knowing that $x+y+z$ and $x^{3}+y^{3}+z^{3}$ are be real numbers.

$\{a_n\}$ is a positive integer sequence such that $ a_{i+2} = a_{i+1} + a_{i} (i \ge 1) $. For positive integer $n$, define $\{b_n\}$ as \[ b_n = \frac{1}{a_{2n+1}} \sum_{i=1}^{4n-2} { a_i } \] Prove that $b_n$ is positive integer, and find the general form of $b_n$.

Given a sequence of $19$ positive integers not exceeding $88$ and another sequence of $88$ positive integers not exceeding $19$. Show that we can find two subsequences of consecutive terms, one from each sequence, with the same sum.

Find all values of $n$ for which there exists a convex cyclic non-regular polygon with $n$ vertices such that the measures of all its internal angles are equal.

Let $O$ be the centre of a circle and $A$ a fixed interior point of the circle different from $O$. Determine all points $P$ on the circumference of the circle such that the angle $OPA$ is a maximum. [asy] import graph; unitsize(2 cm); pair A, O, P; A = (0.5,0.2); O = (0,0); P = dir(80); draw(Circle(O,1)); draw(O--A--P--cycle); label("$A$", A, E); label("$O$", O, S); label("$P$", P, N); [/asy]

Let $ n>4$ be a positive integer such that $ n$ is composite (not a prime) and divides $ \varphi (n) \sigma (n) \plus{}1$, where $ \varphi (n)$ is the Euler's totient function of $ n$ and $ \sigma (n)$ is the sum of the positive divisors of $ n$. Prove that $ n$ has at least three distinct prime factors.

For each positive integer $n\leq 49$ we define the numbers $a_n = 3n+\sqrt{n^2-1}$ and $b_n=2(\sqrt{n^2+n}+\sqrt{n^2-n})$. Prove that there exist two integers $A,B$ such that \[ \sqrt{a_1-b_1}+\sqrt{a_2-b_2} + \cdots + \sqrt{a_{49}-b_{49}} = A+B\sqrt2. \]

Fill in numbers in the boxes below so that the sum of the entries in each three consecutive boxes is $2005$. What is the number that goes into the leftmost box? [asy] size(300); label("999",(2.5,.5)); label("888",(7.5,.5)); draw((0,0)--(9,0)); draw((0,1)--(9,1)); for (int i=0; i<=9; ++i) { draw((i,0)--(i,1)); } [/asy]

The intelligence quotient (IQ) of a country is defined as the average IQ of its entire population. It is assumed that the total population and individual IQs remain constant throughout. (a) (i) A group of people from country $A$ has emigrated to country $B$ . Show that it can happen that as a result , the IQs of both countries have increased. (ii) After this, a group of people from $B$, which may include immigrants from $A$, emigrates to $A$. Can it happen that the IQs of both countries will increase again? (b) A group of people from country $A$ has emigrated to country $B$, and a group of people from $B$ has emigrated to country $C$ . It is known that a s a result , the IQs o f all three countries have increased. After this, a group of people from $C$ emigrates to $B$ and a group of people from $B$ emigrates to $A$. Can it happen that the IQs of all three countries will increase again? (A Kanel, B Begun)

In the coordinate plane,the graps of functions $y=sin x$ and $y=tan x$ are drawn, along with the coordinate axes. Using compass and ruler, construct a line tangent to the graph of sine at a point above the axis, $Ox$, as well at a point below that axis (the line can also meet the graph at several other points)

Prove that there does not exist a natural number which, upon transfer of its initial digit to the end, is increased five, six or eight times.