Problems

Prove that if the numbers $x,y,z$ satisfy the following system of equations for some values of $p$ and $q$: $$\begin{array}{rl}y& =x^2+px+q,\\ z& =y^2+py+q,\\ x& =z^2+pz+q,\end{array}$$ then the inequality $x^{2}y+y^{2}z+z^{2}x\ge x^{2}z+y^{2}x+z^{2}y$ is satisfied.

Can I replace the letters with numbers in the puzzle $RE\times CTS+1=CE\times MS$ so that the correct equality is obtained (different letters need to be replaced by different numbers, and the same letters must correspond to the same digits)?

Find $x^3+y^3$ if $x+y=5$ and $x+y+x^{2}y+x{y}^2=24$.

On the $xy$-plane shown below is the graph of the function $y=a{x}^2+c$. At which points does the graph of the function $y=cx+a$ intersect the $x$ and $y$ axes?

Find the largest natural number $n$ which satisfies $n^{200}<5^{300}$.

Does there exist a natural number which, when divided by the sum of its digits, gives a quotient and remainder both equal to the number 2011?

Along the path between Fiona’s and Jane’s house there is a row of flowers: 15 peonies and 15 tulips in a random order. Before visiting Fiona’s house, Jane started watering all of the plants from the beginning of the row. After the 10th tulip the water finished and 10 flowers were left unwatered. The next day, before visiting Jane’s house, Fiona started picking flowers for a bouquet starting from the end of the row. After picking the 6th tulip, she decided that the bouquet was big enough. How many flowers were left growing beside the path?

Three circles are constructed on a triangle, with the medians of the triangle forming the diameters of the circles. It is known that each pair of circles intersects. Let $C_1$ be the point of intersection, further from the vertex $C$, of the circles constructed from the medians $A{M}_1$ and $B{M}_2$. Points $A_1$ and $B_1$ are defined similarly. Prove that the lines $A{A}_1$, $B{B}_1$ and $C{C}_1$ intersect at the same point.

In a herd consisting of horses and camels (some with one hump and some with two) there are a total of 200 humps. How many animals are in the herd, if the number of horses is equal to the number of camels with two humps?

Of the four inequalities $2x>70$, $x<100$, $4x>25$ and $x>5$, two are true and two are false. Find the value of $x$ if it is known that it is an integer.