Mr. Ramos gave a test to his class of $20$ students. The dot plot below shows the distribution of test scores. [asy] //diagram by pog . give me 1,000,000,000 dollars for this diagram size(5cm); defaultpen(0.7); dot((0.5,1)); dot((0.5,1.5)); dot((1.5,1)); dot((1.5,1.5)); dot((2.5,1)); dot((2.5,1.5)); dot((2.5,2)); dot((2.5,2.5)); dot((3.5,1)); dot((3.5,1.5)); dot((3.5,2)); dot((3.5,2.5)); dot((3.5,3)); dot((4.5,1)); dot((4.5,1.5)); dot((5.5,1)); dot((5.5,1.5)); dot((5.5,2)); dot((6.5,1)); dot((7.5,1)); draw((0,0.5)--(8,0.5),linewidth(0.7)); defaultpen(fontsize(10.5pt)); label("$65$", (0.5,-0.1)); label("$70$", (1.5,-0.1)); label("$75$", (2.5,-0.1)); label("$80$", (3.5,-0.1)); label("$85$", (4.5,-0.1)); label("$90$", (5.5,-0.1)); label("$95$", (6.5,-0.1)); label("$100$", (7.5,-0.1)); [/asy] Later Mr. Ramos discovered that there was a scoring error on one of the questions. He regraded the tests, awarding some of the students $5$ extra points, which increased the median test score to $85$. What is the minimum number of students who received extra points? (Note that the [i]median[/i] test score equals the average of the $2$ scores in the middle if the $20$ test scores are arranged in increasing order.) $\textbf{(A)} ~2\qquad\textbf{(B)} ~3\qquad\textbf{(C)} ~4\qquad\textbf{(D)} ~5\qquad\textbf{(E)} ~6\qquad$

Let $ABCD$ be a convex quadrilateral and let $P$ and $Q$ be variable points inside this quadrilateral so that $\angle APB=\angle CPD=\angle AQB=\angle CQD$. Prove that the lines $PQ$ obtained in this way all pass through a fixed point , or they are all parallel.

Let $m$ be a positive integer. Define the sequence $a_0, a_1, a_2, \cdots$ by $a_0 = 0,\; a_1 = m,$ and $a_{n+1} = m^2a_n - a_{n-1}$ for $n = 1,2,3,\cdots$. Prove that an ordered pair $(a,b)$ of non-negative integers, with $a \leq b$, gives a solution to the equation \[ {\displaystyle \frac{a^2 + b^2}{ab + 1} = m^2} \] if and only if $(a,b)$ is of the form $(a_n,a_{n+1})$ for some $n \geq 0$.

Let $\alpha , \beta , \gamma , \delta$ be positive numbers such that for all $x$, $\sin{\alpha x}+\sin {\beta x}=\sin {\gamma x}+\sin {\delta x}$. Prove that $\alpha =\gamma$ or $\alpha=\delta$.

Find all polynomials $P(x), Q(x)$ with real coefficients such that for all real numbers $a$, $P(a)$ is a root of the equation $x^{2023}+Q(a)x^2+(a^{2024}+a)x+a^3+2025a=0.$

Solve the equation: $$\frac{x^2}{3}+\frac{48}{x^3}=10 \left(\frac{x}{3}-\frac{4 }{x} \right)$$

Consider six arbitrary points in space. Every two points are joined by a segment. Prove that there are two triangles that can not be separated. [img]http://i38.tinypic.com/35n615y.png[/img]

For each positive integer $n$, let $f(n)$ be the maximal natural number such that: $2^{f(n)}$ divides $\sum^{\left\lfloor \frac{n - 1}{2}\right\rfloor}_{i=0} \binom{n}{2 \cdot i + 1} 3^i$. Find all $n$ such that $f(n) = 1996.$ [hide="old version"]For each positive integer $n$, let $f(n)$ be the maximal natural number such that: $2^{f(n)}$ divides $\sum^{n + 1/2}_{i=1} \binom{2 \cdot i + 1}{n}$. Find all $n$ such that $f(n) = 1996.$[/hide]

In a triangle $ABC,$ incircle touches the sides $BC, CA, AB$ at $D, E, F,$ respectively. A circle $\omega$ passing through $A$ and tangent to line $BC$ at $D$ intersects the line segments $BF$ and $CE$ at $K$ and $L,$ respectively. The line passing through $E$ and parallel to $DL$ intersects the line passing through $F$ and parallel to $DK$ at $P.$ If $R_1, R_2, R_3, R_4$ denotes the circumradius of the triangles $AFD, AED, FPD, EPD,$ respectively, prove that $R_1R_4=R_2R_3.$

In a scalene triangle $\Delta ABC$, $AB<AC$, $PB$ and $PC$ are tangents of the circumcircle $(O)$ of $\Delta ABC$. A point $R$ lies on the arc $\widehat{AC}$(not containing $B$), $PR$ intersects $(O)$ again at $Q$. Suppose $I$ is the incenter of $\Delta ABC$, $ID \perp BC$ at $D$, $QD$ intersects $(O)$ again at $G$. A line passing through $I$ and perpendicular to $AI$ intersects $AB,AC$ at $M,N$, respectively. Prove that, if $AR \parallel BC$, then $A,G,M,N$ are concyclic.

Find all polynomials with degree $\leq n$ and nonnegative coefficients, such that $P(x)P(\frac{1}{x}) \leq P(1)^2$ for every positive $x$

Form a square with sides of length $5$, triangular pieces from the four coreners are removed to form a regular octagonn. Find the area [b]removed[/b] to the nearest integer.

The sequence $(x_n)$ is given by $x_1=\frac{1}{2},$ $x_n=\frac{2n-3}{2n} \cdot x_{n-1}$ for $n=2,3,... .$ Prove that for all natural numbers $n \geq 1$ the following inequality holds $x_1+x_2+...+x_n < 1$.

Let each of $m$ distinct points on the positive part of the $x$-axis be joined to $n$ distinct points on the positive part of the $y$-axis. Obtain a formula for the number of intersection points of these segments, assuming that no three of the segments are concurrent.

Does there exist functions $ f,g: \mathbb{R}\to\mathbb{R}$ such that $ f(g(x)) \equal{} x^2$ and $ g(f(x)) \equal{} x^k$ for all real numbers $ x$ a) if $ k \equal{} 3$? b) if $ k \equal{} 4$?

Find $x$ so that $2^{2^{3^{2^{2}}}} = 4^{4^{x}}$.

In triangle $ABC$, $AB > BC > CA$, $AB=6$, $\angle{B}-\angle{C}=90^o$. The incircle touches $BC$ at $E$ and $EF$ is a diameter of the incircle. Radical $AF$ intersect $BC$ at $D$. $DE$ equals to the circumradius of $\triangle{ABC}$. Find $BC$ and $AC$.

Given a positive integer $n$ and $I = \{1, 2,..., k\}$ with $k$ is a positive integer. Given positive integers $a_1, a_2, ..., a_k$ such that for all $i \in I$: $1 \le a_i \le n$ and $$\sum_{i=1}^k a_i \ge 2(n!).$$ Show that there exists $J \subseteq I$ such that $$n! + 1 \ge \sum_{j \in J}a_j >\sqrt {n! + (n - 1)n}$$

Let be $ a,b,c $ positive integers.Prove that $ |a-b\sqrt{c}|<\frac{1}{2b} $ is true if and only if $ |a^{2}-b^{2}c|<\sqrt{c} $.

$ABCD$ is a convex quadrilateral, and the point $K$ is a point on side $AB$, where $\angle KDA=\angle BCD$, let $L$ be a point on the diagonal $AC$, where $KL$ is parallel to $BC$. Prove that $$\angle KDB=\angle LDC.$$ [i](tatari/nightmare)[/i]

In a convex quadrilateral $ABCD$, $\angle DAB=\angle BCD$ and the angle bisector of $\angle ABC$ passes through the middle point of $CD$. If $CD=3AD$, determine the ratio $\frac{AB}{BC}$.

$ABC$-triangle with circumcenter $O$ and $\angle B=30$. $BO$ intersect $AC$ at $K$. $L$ - midpoint of arc $OC$ of circumcircle $KOC$, that does not contains $K$. Prove, that $A,B,L,K$ are concyclic.

Find the sum of the squares of all values of $x$ that satisfy $\log_2 (x + 3) + \log_2 (2 - x) = 2$.

Given a prime $p$ and a real number $\lambda \in (0,1)$. Let $s$ and $t$ be positive integers such that $s \leqslant t < \frac{\lambda p}{12}$. $S$ and $T$ are sets of $s$ and $t$ consecutive positive integers respectively, which satisfy $$\left| \left\{ (x,y) \in S \times T : kx \equiv y \pmod p \right\}\right| \geqslant 1 + \lambda s.$$Prove that there exists integers $a$ and $b$ that $1 \leqslant a \leqslant \frac{1}{ \lambda}$, $\left| b \right| \leqslant \frac{t}{\lambda s}$ and $ka \equiv b \pmod p$.

Let $n\geq 3$ be integer. Given two pairs of $n$ cards numbered from 1 to $n$. Mix the $2n$ cards up and take the card 3 times every one card. Denote $X_1,\ X_2,\ X_3$ the numbers of the cards taken out in this order taken the cards. Find the probabilty such that $X_1<X_2<X_3$. Note that once a card taken out, it is not taken a back.