Randy drove the first third of his trip on a gravel road, the next 20 miles on pavement, and the remaining one-fifth on a dirt road. In miles, how long was Randy's trip? $ \textbf {(A) } 30 \qquad \textbf {(B) } \frac{400}{11} \qquad \textbf {(C) } \frac{75}{2} \qquad \textbf {(D) } 40 \qquad \textbf {(E) } \frac{300}{7} $

For which $ p$ prime numbers, there is an integer root of the polynominal $ 1 \plus{} p \plus{} Q(x^1)\cdot\ Q(x^2)\ldots\ Q(x^{2p \minus{} 2})$ such that $ Q(x)$ is a polynominal with integer coefficients?

Find the largest value of the area of the projection of the cylinder onto the plane if its radius is $r$ and its height is $h$ (orthogonal projection).

Let $a,b$ and $c$ be positive integers, no two of which have a common divisor greater than $1$. Show that $2abc-ab-bc-ca$ is the largest integer which cannot be expressed in the form $xbc+yca+zab$, where $x,y,z$ are non-negative integers.

Evaluate \[ \int_2^4 \frac{\sqrt{\ln(9-x)}\,dx}{\sqrt{\ln(9-x)}+\sqrt{\ln(x+3)}}. \]

Let $f(n)$ be the sum of the factors of $2^n \cdot 31.$ Find $\sum_{n=0}^{4} f(n).$

When a right triangle is rotated about one leg, the volume of the cone produced is $800 \pi$ $\text{cm}^3$. When the triangle is rotated about the other leg, the volume of the cone produced is $1920 \pi$ $\text{cm}^3$. What is the length (in cm) of the hypotenuse of the triangle?

Let $ABCD$ be a convex quadrilateral and $K$ be the common point of rays $AB$ and $DC$. There exists a point $P$ on the bisectrix of angle $AKD$ such that lines $BP$ and $CP$ bisect segments $AC$ and $BD$ respectively. Prove that $AB = CD$.

Consider the rectangle $ ABCD $ and the points $ M,N,P,Q $ on the segments $ AB,BC,CD, $ respectively, $ DA, $ excluding its extremities. Denote with $ p_{\square} , A_{\square} $ the perimeter, respectively, the area of $ \square. $ Prove that: [b]a)[/b] $ p_{MNPQ}\ge AC+BD. $ [b]b)[/b] $ p_{MNPQ} =AC+BD\implies A_{MNPQ}\le \frac{A_{ABCD}}{2} . $ [b]c)[/b] $ p_{MNPQ} =AC+BD\implies MP^2 +NQ^2\ge AC^2. $ [i]Dan Brânzei[/i] and [i]Gheorghe Iurea[/i]

Suppose that $ f(x),\ g(x)$ are differential fuctions and their derivatives are continuous. Find $ f(x),\ g(x)$ such that $ f(x)\equal{}\frac 12\minus{}\int_0^x \{f'(t)\plus{}g(t)\}\ dt\ \ g(x)\equal{}\sin x\minus{}\int_0^{\pi} \{f(t)\minus{}g'(t)\}\ dt$.

given any 3 distinct points $X,Y,Z$on the integer coordinates of the x-axis,the following operation is allowed:A point say $X$ is reflected over another point say $Y$. Note that after each operation only one among three points is moved. we perform these operations till 2 out of the 3 points coincide. let $N=N(X,Y,Z)$ denote the minimum number of operations before we are forced to stop.(this could happen in different ways). show that there are at most $2^N$coordinates that point $X$ could end up if we are forced to stop after $N$operations

A four - digit natural number which is divisible by $7$ is given. The number obtained by writing the digits in reverse order is also divisible by $7$. Furthermore, both the numbers leave the same remainder when divided by $37$. Find the 4-digit number.

Triangle $ABC$ has $AB=4,BC=6,CA=5$. Let $M$ be the midpoint of $\overline{BC}$ and $P$ the point on the circumcircle of $\triangle ABC$ such that $\angle MPA=90^\circ$. Let points $D$ and $E$ lie on $\overline{AC}$ and $\overline{AB}$ respectively such that $\overline{BD}\perp\overline{AC}$ and $\overline{CE}\perp\overline{AB}$. Find $\tfrac{PD}{PE}$.

Prove that if coefficients of the quadratic equation $ ax^2\plus{}bx\plus{}c\equal{}0$ are odd integers, then the roots of the equation cannot be rational numbers.

In a unit cube, $CG$ is the edge perpendicular to the face $ABCD$. Let $O_1$ be the incircle of square $ABCD$ and $O_2$ be the circumcircle of triangle $BDG$. Determine min$\{XY|X\in O_1,Y\in O_2\}$.

There are $2019$ cities in the country of Balticwayland. Some pairs of cities are connected by non-intersecting bidirectional roads, each road connecting exactly 2 cities. It is known that for every pair of cities $A$ and $B$ it is possible to drive from $A$ to $B$ using at most $2$ roads. There are $62$ cops trying to catch a robber. The cops and robber all know each others’ locations at all times. Each night, the robber can choose to stay in her current city or move to a neighbouring city via a direct road. Each day, each cop has the same choice of staying or moving, and they coordinate their actions. The robber is caught if she is in the same city as a cop at any time. Prove that the cops can always catch the robber

On the sides $BC,CA$ and $AB$ of a triangle $ABC$ of area $S$ are taken points $A' ,B' ,C'$ respectively such that $AC' /AB = BA' /BC = CB' /CA = p$, where $0 < p < 1$ is variable. (a) Find the area of triangle $A' B' C'$ in terms of $ p$. (b) Find the value of $p$ which minimizes this area. (c) Find the locus of the intersection point $P$ of the lines through $A' $ and $C'$ parallel to $AB$ and $AC$ respectively.

Find all pairs of positive integers $x,y$ such that $$x^3+y^3=4(x^2y+xy^2-5).$$

Find all the functions $f(x),$ continuous on the whole real axis, such that for every real $x$ \[f(3x-2)\leq f(x)\leq f(2x-1).\] [i]Proposed by A. Golovanov[/i]

Let be a continuous function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ that has the property that $$ xf(x)\ge \int_0^x f(t)dt , $$ for all real numbers $ x. $ Prove that [b]a)[/b] the mapping $ x\mapsto \frac{1}{x}\int_0^x f(t) dt $ is nondecreasing on the restrictions $ \mathbb{R}_{<0 } $ and $ \mathbb{R}_{>0 } . $ [b]b)[/b] if $ \int_x^{x+1} f(t)dt=\int_{x-1}^x f(t)dt , $ for any real number $ x, $ then $ f $ is constant. [i]Mihai Piticari[/i]

In a table consisting of $2021\times 2021$ unit squares, some unit squares are colored black in such a way that if we place a mouse in the center of any square on the table it can walk in a straight line (up, down, left or right along a column or row) and leave the table without walking on any black square (other than the initial one if it is black). What is the maximum number of squares that can be colored black?

Let $A$, $B$, $C$, $D$, $E$, $F$ be six points on a circle such that $AE\parallel BD$ and $BC\parallel DF$. Let $X$ be the reflection of the point $D$ in the line $CE$. Prove that the distance from the point $X$ to the line $EF$ equals to the distance from the point $B$ to the line $AC$.

We will call one of the cells of a rectangle 11 x 13 “[i]peculiar[/i]” , if after removing it the remaining figure can be cut into squares 2 x 2 and 3 x 3. How many of the 143 cells are “[i]peculiar[/i]”?

Let $a$ and $b$ be two positive integers. Prove that the integer \[a^2+\left\lceil\frac{4a^2}b\right\rceil\] is not a square. (Here $\lceil z\rceil$ denotes the least integer greater than or equal to $z$.) [i]Russia[/i]

Let $x,y$ be the positive real numbers with $x+y=1$, and $n$ be the positive integer with $n\ge2$. Prove that \[\frac{x^n}{x+y^3}+\frac{y^n}{x^3+y}\ge\frac{2^{4-n}}{5}\]