MathDB

1

Part of 2017 Iran MO (3rd round)

Problems(8)

Polynomials

Source: Iran 3rd round 2017 first Algebra exam

8/7/2017
Find all polynomials P(x)P(x) and Q(x)Q(x) with real coefficients such that P(Q(x))=P(x)2017P(Q(x))=P(x)^{2017} for all real numbers xx.
algebrapolynomialIranIranMO
Number Theory : primes

Source: Iran 3rd round 2017 Number theory first exam-P1

8/9/2017
Let nn be a positive integer. Consider prime numbers p1,…,pkp_1,\dots ,p_k. Let a1,…,ama_1,\dots,a_m be all positive integers less than nn such that are not divisible by pip_i for all 1≤i≤n1 \le i \le n. Prove that if m≥2m\ge 2 then 1a1+⋯+1am\frac{1}{a_1}+\dots+\frac{1}{a_m} is not an integer.
number theoryDivisibilityprime numbersprimeIranIranMO
Iran geometry

Source: Iran MO 3rd round 2017 mid-terms - Geometry P1

8/10/2017
Let ABCABC be a triangle. Suppose that X,YX,Y are points in the plane such that BX,CYBX,CY are tangent to the circumcircle of ABCABC, AB=BX,AC=CYAB=BX,AC=CY and X,Y,AX,Y,A are in the same side of BCBC. If II be the incenter of ABCABC prove that ∠BAC+∠XIY=180\angle BAC+\angle XIY=180.
geometrycircumcircleincenter
P1- Combinatorics from 2017 Iran MO 3rd round

Source: 2017 Iran MO 3rd round, first combinatorics exam P1

9/12/2017
There's a tape with n2n^2 cells labeled by 1,2,…,n21,2,\ldots,n^2. Suppose that x,yx,y are two distinct positive integers less than or equal to nn. We want to color the cells of the tape such that any two cells with label difference of xx or yy have different colors. Find the minimum number of colors needed to do so.
Irancombinatorics
Iran Geometry

Source: Iran MO 3rd round 2017 finals - Geometry P1

9/3/2017
Let ABCABC be a right-angled triangle (∠A=90∘)\left(\angle A=90^{\circ}\right) and MM be the midpoint of BCBC. ω1\omega_1 is a circle which passes through B,MB,M and touchs ACAC at XX. ω2\omega_2 is a circle which passes through C,MC,M and touchs ABAB at YY (X,YX,Y and AA are in the same side of BCBC). Prove that XYXY passes through the midpoint of arc BCBC (does not contain AA) of the circumcircle of ABCABC.
geometrycircumcircle
Numbers Theory

Source: Iran 3rd round 2017 Numbers theory final exam-P1

8/30/2017
Let xx and yy be integers and let pp be a prime number. Suppose that there exist realatively prime positive integers mm and nn such that xm≡yn(modp)x^m \equiv y^n \pmod p Prove that there exists an unique integer zz modulo pp such that x \equiv z^n \pmod p   \text{and}   y \equiv z^m \pmod p
number theoryIran 3rd Round
Functional equation

Source: Iran 3rd round-2017-Algebra final exam-P1

9/2/2017
Let R≥0\mathbb{R}^{\ge 0} be the set of all nonnegative real numbers. Find all functions f:R≥0→R≥0f:\mathbb{R}^{\ge 0} \to \mathbb{R}^{\ge 0} such that x+2max⁡{y,f(x),f(z)}≥f(f(x))+2max⁡{z,f(y)} x+2 \max\{y,f(x),f(z)\} \ge f(f(x))+2 \max\{z,f(y)\} for all nonnegative real numbers x,yx,y and zz.
algebrafunctional equation
P1- Second combinatorics exam of 2017 Iran MO 3rd round

Source: 2017 Iran MO 3rd round , second combinatorics exam P1

9/12/2017
There are 100100 points on the circumference of a circle, arbitrarily labelled by 1,2,…,1001,2,\ldots,100. For each three points, call their triangle clockwise if the increasing order of them is in clockwise order. Prove that it is impossible to have exactly 20172017 clockwise triangles.
Irancombinatorics