Let $r,s,u,v$ be real numbers. Prove that: $$min\{r-s^2,s-u^2, u-v^2,v-r^2\}\le \frac{1}{4}$$

Let $PA_1A_2...A_{12} $ be the regular pyramid, $ A_1A_2...A_{12} $ is regular polygon, $S$ is area of the triangle $PA_1A_5$ and angle between of the planes $A_1A_2...A_{12} $ and $ PA_1A_5 $ is equal to $ \alpha $. Find the volume of the pyramid.

In the adjacent figure, can the numbers $1,2,3, 4,..., 18$ be placed, one on each line segment, such that the sum of the numbers on the three line segments meeting at each point is divisible by $3$?

Find all triples $(a, b, c)$ of positive integers with $a+b+c = 10$ such that there are $a$ red, $b$ blue and $c$ green points (all different) in the plane satisfying the following properties: $\bullet$ for each red point and each blue point we consider the distance between these two points, the sum of these distances is $37$, $\bullet$ for each green point and each red point we consider the distance between these two points, the sum of these distances is $30$, $\bullet$ for each blue point and each green point we consider the distance between these two points, the sum of these distances is $1$.

Let $ABC$ be a right-angled triangle with $\angle A = 90^{\circ}$. Let $K$ be the midpoint of $BC$, and let $AKLM$ be a parallelogram with centre $C$. Let $T$ be the intersection of the line $AC$ and the perpendicular bisector of $BM$. Let $\omega_1$ be the circle with centre $C$ and radius $CA$ and let $\omega_2$ be the circle with centre $T$ and radius $TB$. Prove that one of the points of intersection of $\omega_1$ and $\omega_2$ is on the line $LM$. [i]Proposed by Greece[/i]

Is it possible to put distinct positive integers less than $1991$ in the cells of a $9\times 9$ table so that the products of all the numbers in every column and every row are equal to each other? (N.B. Vasiliev, Moscow)

A ray is drawn from the origin tangent to the graph of the upper part of the hyperbola $y^2=x^2-x+1$ in the first quadrant. This ray makes an angle of $\theta$ with the positive $x$-axis. Compute $\cos\theta$.

A positive integer is called [i]uphill [/i] if the digits in its decimal representation form an increasing sequence from left to right. That is, a number $\overline{a_1a_2... a_n}$ is uphill if $a_i \le a_{i+1}$ for all $i$. For example, $123$ and $114$ are both uphill. Suppose a polynomial $P(x)$ with rational coefficients takes on an integer value for each uphill positive integer $x$. Is it necessarily true that $P(x)$ takes on an integer value for each integer $x$?

Let $ABC$ be an isosceles triangle such that $AB = AC$ and $\angle A = 20^o$. Let $M$ be the foot of the altitude from $C$ and let $N$ be a point on the side $AC$ such that $CN =\frac12 BC$. Determine the measure of the angle $AMN$.

Let the circle $\Omega$ and the line $\ell$ intersect at two different points $A{}$ and $B{}$. For different and non-points. Let $X$ and $T$ be points on $\ell$ and $Y$ and $Z$ be points on $\Omega$, all of them different from $A{}$ and $B{}$. Prove the following statements: [list=a] [*]The points $X,Y$ and $Z$ lie on the same line if and only if \[\frac{\overline{AX}}{\overline{BX}}=\pm\frac{AY}{BY}\cdot\frac{AZ}{BZ}.\] [*]The points $X,Y,Z$ and $T$ lie on the same circle if and only if \[\frac{\overline{AX}}{\overline{BX}}\cdot\frac{\overline{AT}}{\overline{BT}}=\pm\frac{AY}{BY}\cdot\frac{AZ}{BZ}.\] [/list] Note: In both points, the sign $+$ is selected in the right parts of the equalities if the points $Y{}$ and $Z{}$ lie on the same arc $AB$ of the circle $\Omega$, and the sign $-$ if $Y{}$ and $Z{}$ lie on different arcs $AB$. By $\overline{AX}/\overline{BX}$, we indicate the ratio of the lengths of $AX$ and $BX$, taken with the sign $+$ or $-$ depending on whether the $AX$ and $BX$ vectors are co-directed or oppositely directed. [i]Proposed by M. Skopenkov[/i]

The average age of the participants in a mathematics competition (gymnasts and high school students) increases by exactly one month if three high school age students $18$ years each are included in the competition or if three gymnasts aged $12$ years each are excluded from the competition. How many participants were initially in the contest?

A sequence of natural numbers is constructed by listing the first $4$, then skipping one, listing the next $5$, skipping $2$, listing $6$, skipping $3$, and, on the $n$th iteration, listing $n+3$ and skipping $n$. The sequence begins $1,2,3,4,6,7,8,9,10,13$. What is the $500,000$th number in the sequence? $ \textbf{(A)}\ 996,506\qquad\textbf{(B)}\ 996507\qquad\textbf{(C)}\ 996508\qquad\textbf{(D)}\ 996509\qquad\textbf{(E)}\ 996510 $

Simon and Garfunkle play in a round-robin golf tournament. Each player is awarded one point for a victory, a half point for a tie, and no points for a loss. Simon beat Garfunkle in the first game by a record margin as Garfunkle sent a shot over the bridge and into troubled waters on the final hole. Garfunkle went on to score $8$ total victories, but no ties at all. Meanwhile, Simon wound up with exactly $8$ points, including the point for a victory over Garfunkle. Amazingly, every other player at the tournament scored exactly $n$. Find the sum of all possible values of $n$.

Let \(n \geq 5\) be integer. The convex polygon \(P = A_{1} A_{2} \ldots A_{n}\) is bicentric, that is, it has an inscribed and circumscribed circle. Set \(A_{i+n}=A_{i}\) to every integer \(i\) (that is, all indices are taken modulo \(n\)). Suppose that for all \(i, 1 \leq i \leq n\), the rays \(A_{i-1} A_{i}\) and \(A_{i+2} A_{i+1}\) meet at the point \(B_{i}\). Let \(\omega_{i}\) be the circumcircle of \(B_{i} A_{i} A_{i+1}\). Prove that there is a circle tangent to all \(n\) circles \(\omega_{i}\), \(1 \leq i \leq n\).

Let $ \{k_n\}_{n \equal{} 1}^{\infty}$ be a sequence of real numbers having the properties $ k_1 > 1$ and $ k_1 \plus{} k_2 \plus{} \cdots \plus{} k_n < 2k_n$ for $ n \equal{} 1,2,...$. Prove that there exists a number $ q > 1$ such that $ k_n > q^n$ for every positive integer $ n$.

Let $ABC$ be a triangle. Let $ABC_1, BCA_1, CAB_1$ be three equilateral triangles that do not overlap with $ABC$. Let $P$ be the intersection of the circumcircles of triangle $ABC_1$ and $CAB_1$. Let $Q$ be the point on the circumcircle of triangle $CAB_1$ so that $PQ$ is parallel to $BA_1$. Let $R$ be the point on the circumcircle of triangle $ABC_1$ so that $PR$ is parallel to $CA_1$. Show that the line connecting the centroid of triangle $ABC$ and the centroid of triangle $PQR$ is parallel to $BC$. [i]Proposed by usjl[/i]