A small ship sails on an infinite coordinate sea. At the moment $t$ the ship is at the point with coordinates $(f(t), g(t))$, where $f$ and $g$ are two polynomials of third degree. Yesterday at $14:00$ the ship was at the same point as at $13:00$, and at $20:00$, it was at the same point as at $19:00$. Prove that the ship sails along a straight line.

Let $K$ be a number field which is neither $\mathbb{Q}$ nor a quadratic imaginary extension of $\mathbb{Q}$. Denote by $\mathcal{L}(K)$ the set of integers $n\ge 3$ for which we can find units $\varepsilon_1,\ldots,\varepsilon_n\in K$ for which $$\varepsilon_1+\dots+\varepsilon_n=0,$$but $\displaystyle\sum_{i\in I}\varepsilon_i\neq 0$ for any nonempty proper subset $I$ of $\{1,2,\dots,n\}$. Prove that $\mathcal{L}(K)$ is infinite, and that its smallest element can be bounded from above by a function of the degree and discriminant of $K$. Further, show that for infinitely many $K$, $\mathcal{L}(K)$ contains infinitely many even and infinitely many odd elements.

Prove that a pyramid $A_1A_2 \ldots A_{2k+1}S$ with equal lateral edges and equal space angles between adjacent lateral walls is regular.

How many positive integers less than or equal to $1000$ are divisible by $2$ and $3$ but not by $5$? [i]2015 CCA Math Bonanza Individual Round #6[/i]

Given a scalene triangle $ABC$ with $\angle B=130^{\circ}$. Let $H$ be the foot of altitude from $B$. $D$ and $E$ are points on the sides $AB$ and $BC$, respectively, such that $DH=EH$ and $ADEC$ is a cyclic quadrilateral. Find $\angle{DHE}$.

In the plane let $\,C\,$ be a circle, $\,L\,$ a line tangent to the circle $\,C,\,$ and $\,M\,$ a point on $\,L$. Find the locus of all points $\,P\,$ with the following property: there exists two points $\,Q,R\,$ on $\,L\,$ such that $\,M\,$ is the midpoint of $\,QR\,$ and $\,C\,$ is the inscribed circle of triangle $\,PQR$.

Let $x, y, z$ be three real distinct numbers such that $$\begin{cases} x^2-x=yz \\ y^2-y=zx \\ z^2-z=xy \end{cases}$$ Show that $-\frac{1}{3} < x,y,z < 1$.

Let $n$ be a positive integer and $p$ a prime number $p > n + 1$. Prove that the following equation does not have integer solution $$1 + \frac{x}{n + 1} + \frac{x^2}{2n + 1} + ...+ \frac{x^p}{pn + 1} = 0$$ Luu Ba Thang, Department of Mathematics, College of Education

Let $P(z)= z^n + c_1 z^{n-1} + c_2 z^{n-2} + \cdots + c_n$ be a polynomial in the complex variable $z$, with real coefficients $c_k$. Suppose that $|P(i)| < 1$. Prove that there exist real numbers $a$ and $b$ such that $P(a + bi) = 0$ and $(a^2 + b^2 + 1)^2 < 4 b^2 + 1$.

How many permutations of the set $\{B, M, T, 2,0\}$ do not have $B$ as their first element?

Given a $\triangle ABC$ with sides $a,b,c.$ Three tangents are drawn to the incircle of $\triangle ABC$ parallel to the sides of $\triangle ABC$.These tangents cut [b]three new little triangles[/b].Three little incircles are drawn into new little triangles.[b][u]Find the sum of the area of these 4 incircles.[/u][/b]

Given $n$ points in the plane, show that we can always find three which give an angle $\le \pi / n$.

A four-digit positive integer is called [i]virtual[/i] if it has the form $\overline{abab}$, where $a$ and $b$ are digits and $a \neq 0$. For example 2020, 2121 and 2222 are virtual numbers, while 2002 and 0202 are not. Find all virtual numbers of the form $n^2+1$, for some positive integer $n$.

How many sequences $a_1$, $a_2$, $...$,$a_8$ of zeroes and ones have $a_1a_2 + a_2a_3 +...+ a_7a_8 = 5$?

In a convex quadrilateral $ABCD$, $\angle ABC = \angle CDA$. Let $X \neq C$ be the intersection of the circumcircle of $\triangle BCD$ and circle with diameter $AC$. Prove that the tangent to the circumcircle of $\triangle BCD$ at $X$, the tangent to the circumcircle of $\triangle ABD$ at $A$ concur on $BD$.

Let $m,n \geqslant 2$ be integers, let $X$ be a set with $n$ elements, and let $X_1,X_2,\ldots,X_m$ be pairwise distinct non-empty, not necessary disjoint subset of $X$. A function $f \colon X \to \{1,2,\ldots,n+1\}$ is called [i]nice[/i] if there exists an index $k$ such that \[\sum_{x \in X_k} f(x)>\sum_{x \in X_i} f(x) \quad \text{for all } i \ne k.\] Prove that the number of nice functions is at least $n^n$.

In a football tournament with $n \geqslant 4$ teams, each pair of teams played against each other exactly once. In the final table, the scores of the teams are $n$ consecutive numbers. Find the maximum possible score of the winner of the tournament. [b]Note:[/b] A victory gives $3$ points, a draw gives $1$ point and a loss gives $0$ points.

There are three piles of coins, with $a,b$ and $c$ coins respectively, where $a,b,c\geq2015$ are positive integers. The following operations are allowed: (1) Choose a pile with an even number of coins and remove all coins from this pile. Add coins to each of the remaining two piles with amount equal to half of that removed; or (2) Choose a pile with an odd number of coins and at least 2017 coins. Remove 2017 coins from this pile. Add 1009 coins to each of the remaining two piles. Suppose there are sufficiently many spare coins. Find all ordered triples $(a,b,c)$ such that after some finite sequence of allowed operations. There exists a pile with at least $2017^{2017}$ coins.

Let $k_1, k_2$ and $k_3$ be three circles with centers $O_1, O_2$ and $O_3$ respectively, such that no center is inside of the other two circles. Circles $k_1$ and $k_2$ intersect at $A$ and $P$, circles $k_1$ and $k_3$ intersect and $C$ and $P$, circles $k_2$ and $k_3$ intersect at $B$ and $P$. Let $X$ be a point on $k_1$ such that the line $XA$ intersects $k_2$ at $Y$ and the line $XC$ intersects $k_3$ at $Z$, such that $Y$ is nor inside $k_1$ nor inside $k_3$ and $Z$ is nor inside $k_1$ nor inside $k_2$. a) Prove that $\triangle XYZ$ is simular to $\triangle O_1O_2O_3$ b) Prove that the $P_{\triangle XYZ} \le 4P_{\triangle O_1O_2O_3}$. Is it possible to reach equation?$ *Note: $P$ denotes the area of a triangle*

The diagonals \( AD \), \( BE \), and \( CF \) of a hexagon \( ABCDEF \) inscribed in a circle \( k \) intersect at a point \( P \), and the acute angle between any two of them is \( 60^\circ \). Let \( r_{AB} \) be the radius of the circle tangent to segments \( PA \) and \( PB \) and internally tangent to \( k \); the radii \( r_{BC} \), \( r_{CD} \), \( r_{DE} \), \( r_{EF} \), and \( r_{FA} \) are defined similarly. Prove that \[ r_{AB}r_{CD} + r_{CD}r_{EF} + r_{EF}r_{AB} = r_{BC}r_{DE} + r_{DE}r_{FA} + r_{FA}r_{BC}. \]

A square of side $a$ is inscribed in a triangle so that two of the square’s vertices lie on the base, and the other two lie on the sides of the triangle. Prove that if $r$ is the radius of the circle inscribed in the triangle, then $r\sqrt2 < a < 2r$.

A circle $\omega$ with center $I$ is located inside the circle $\Omega$ with center $O$. Ray $IO$ intersects $\omega$ and $\Omega$ at $P_1$ and $P_2$ respectively. On $\Omega$ an arbitrary point $A \neq P_2$ is chosen. The circumcircle of the triangle $P_1P_2A$ intersects $\omega$ for the second time at $X$. Line $AX$ intersects $\Omega$ for the second time at $Y$. Prove that lines $XP_1$ and $YP_2$ are perpendicular to each other

The positive integers $a,b,c$ are pairwise relatively prime, $a$ and $c$ are odd and the numbers satisfy the equation $a^2+b^2=c^2$. Prove that $b+c$ is the square of an integer.

Let $p$ and $q$ be positive integers such that $p$ is a prime, $p$ divides $q-1$, and $p+q$ divides $p^2+2020q^2$. Find the sum of the possible values of $p$. [i]Proposed by Andy Xu[/i]

Let $ABC$ be a triangle with $AC > BC,$ let $\omega$ be the circumcircle of $\triangle ABC,$ and let $r$ be its radius. Point $P$ is chosen on $\overline{AC}$ such taht $BC=CP,$ and point $S$ is the foot of the perpendicular from $P$ to $\overline{AB}$. Ray $BP$ mets $\omega$ again at $D$. Point $Q$ is chosen on line $SP$ such that $PQ = r$ and $S,P,Q$ lie on a line in that order. Finally, let $E$ be a point satisfying $\overline{AE} \perp \overline{CQ}$ and $\overline{BE} \perp \overline{DQ}$. Prove that $E$ lies on $\omega$.