MathDB

3

Part of 2013 Iran MO (3rd Round)

Problems(5)

a prime number in form of 2x^2+3y^2.

Source: Iran 3rd round 2013 - Number Theory Exam - Problem 3

9/11/2013
Let p>3p>3 a prime number. Prove that there exist x,y∈Zx,y \in \mathbb Z such that p=2x2+3y2p = 2x^2 + 3y^2 if and only if p≡5,11  (mod  24)p \equiv 5, 11 \; (\mod 24) (20 points)
modular arithmeticnumber theory proposednumber theory
Complex numbers!

Source: Iran 3rd round 2013 - Algebra Exam - Problem 3

9/11/2013
For every positive integer n≥2n \geq 2, Prove that there is no n−n-tuple of distinct complex numbers (x1,x2,…,xn)(x_1,x_2,\dots,x_n) such that for each 1≤k≤n1 \leq k \leq n following equality holds. ∏1≤i≤ni≠k(xk−xi)=∏1≤i≤ni≠k(xk+xi)\prod_{\underset{i \neq k}{1 \leq i \leq n}}^{ } (x_k - x_i) = \prod_{\underset{i \neq k}{1 \leq i \leq n}}^{ } (x_k + x_i) (20 points)
complex numbersalgebra proposedalgebra
Race Cars

Source: Iran 3rd round 2013 - Combinatorics exam problem 3

9/18/2014
nn cars are racing. At first they have a particular order. At each moment a car may overtake another car. No two overtaking actions occur at the same time, and except moments a car is passing another, the cars always have an order. A set of overtaking actions is called "small" if any car overtakes at most once. A set of overtaking actions is called "complete" if any car overtakes exactly once. If FF is the set of all possible orders of the cars after a small set of overtaking actions and GG is the set of all possible orders of the cars after a complete set of overtaking actions, prove that ∣F∣=2∣G∣\mid F\mid=2\mid G\mid (20 points) Proposed by Morteza Saghafian
combinatorics unsolvedcombinatorics
Perpendicular Circles Like TST Problem

Source: Iran Third Round 2013 - Geometry Exam - Problem 3

9/8/2013
Suppose line ℓ\ell and four points A,B,C,DA,B,C,D lies on ℓ\ell. Suppose that circles ω1,ω2\omega_1 , \omega_2 passes through A,BA,B and circles ω1′,ω2′\omega'_1 , \omega'_2 passes through C,DC,D. If ω1⊥ω1′\omega_1 \perp \omega'_1 and ω2⊥ω2′\omega_2 \perp \omega'_2 then prove that lines O1O2′,O2O1′,ℓO_1O'_2 , O_2O'_1 , \ell are concurrent where O1,O2,O1′,O2′O_1,O_2,O'_1,O'_2 are center of ω1,ω2,ω1′,ω2′\omega_1 , \omega_2 , \omega'_1 , \omega'_2.
ratiotrigonometrygeometry unsolvedgeometry
Function Generation

Source: Iran 3rd round 2013 - final exam problem 3

9/25/2014
Real function ff generates real function gg if there exists a natural kk such that fk=gf^k=g and we show this by f→gf \rightarrow g. In this question we are trying to find some properties for relation →\rightarrow, for example it's trivial that if f→gf \rightarrow g and g→hg \rightarrow h then f→hf \rightarrow h.(transitivity)
(a) Give an example of two real functions f,gf,g such that f≠gf\not = g ,f→gf\rightarrow g and g→fg\rightarrow f. (b) Prove that for each real function ff there exists a finite number of real functions gg such that f→gf \rightarrow g and g→fg \rightarrow f. (c) Does there exist a real function gg such that no function generates it, except for gg itself? (d) Does there exist a real function which generates both x3x^3 and x5x^5? (e) Prove that if a function generates two polynomials of degree 1 P,QP,Q then there exists a polynomial RR of degree 1 which generates PP and QQ.
Time allowed for this problem was 75 minutes.
functionalgebrapolynomialalgebra unsolved