MathDB

Problem 1

Part of 2004 Croatia National Olympiad

Problems(4)

syseq over reals (Croatian MO 2004 1st Grade P1)

Source:

4/8/2021
Find all real solutions of the system of equations x2−y2=2(xz+yz+x+y),x^2-y^2=2(xz+yz+x+y),y2−z2=2(yx+zx+y+z),y^2-z^2=2(yx+zx+y+z),z2−x2=2(zy+xy+z+x).z^2-x^2=2(zy+xy+z+x).
algebrasystem of equations
areas in pentagon (Croatian MO 2004 2nd Grade P1)

Source:

4/8/2021
Parts of a pentagon have areas x,y,zx,y,z as shown in the picture. Given the area xx, find the areas yy and zz and the area of the entire pentagon. https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvOS9mLzM5NjNjNDcwY2ZmMzgzY2QwYWM0YzI1NmYzOWU2MWY1NTczZmYxLnBuZw==&rn=U2NyZWVuIFNob3QgMjAyMS0wNC0wOCBhdCA0LjMwLjU1IFBNLnBuZw
geometry
possible values of expression in square and point on arc

Source: Croatian MO 2004 3rd Grade P1

4/8/2021
Let ABCDABCD be a square and PP be a point on the shorter arc ABAB of the circumcircle of the square. Which values can the expression AP+BPCP+DP\frac{AP+BP}{CP+DP} take?
geometry
complex inequality

Source: Croatian MO 2004 4th Grade P1

4/9/2021
Let z1,…,znz_1,\ldots,z_n and w1,…,wnw_1,\ldots,w_n (n∈N)(n\in\mathbb N) be complex numbers such that ∣ϵ1z1+…+ϵnzn∣≤∣ϵ1w1+…+ϵnwn∣|\epsilon_1z_1+\ldots+\epsilon_nz_n|\le|\epsilon_1w_1+\ldots+\epsilon_nw_n|holds for every choice of ϵ1,…,ϵn∈{−1,1}\epsilon_1,\ldots,\epsilon_n\in\{-1,1\}. Prove that ∣z1∣2+…+∣zn∣2≤∣w1∣2+…+∣wn∣2.|z_1|^2+\ldots+|z_n|^2\le|w_1|^2+\ldots+|w_n|^2.
InequalityComplex numbercomplex inequalityinequalities