MathDB

1

Part of 2019 Iran MO (3rd Round)

Problems(8)

Inequality

Source: Iran MO 3rd round 2019 mid-terms - Algebra P1

8/1/2019
a,ba,b and cc are positive real numbers so that ∑cyc(a+b)2=2∑cyca+6abc\sum_{\text{cyc}} (a+b)^2=2\sum_{\text{cyc}} a +6abc. Prove that ∑cyc(a−b)2≤∣2∑cyca−6abc∣.\sum_{\text{cyc}} (a-b)^2\leq\left|2\sum_{\text{cyc}} a -6abc\right|.
inequalitiesIran
Planner graphs

Source: Iranian third round 2019 mid term combinatorics exam problem 1

7/27/2019
Hossna is playing with a m∗nm*n grid of points.In each turn she draws segments between points with the following conditions.
**1.** No two segments intersect.
**2.** Each segment is drawn between two consecutive rows.
**3.** There is at most one segment between any two points.
Find the maximum number of regions Hossna can create.
combinatorics
Special quadrilateral

Source: own, Iran MO 3rd round 2019 mid-terms - Geometry P1

7/31/2019
Given a cyclic quadrilateral ABCDABCD. There is a point PP on side BCBC such that ∠PAB=∠PDC=90∘\angle PAB=\angle PDC=90^\circ. The medians of vertexes AA and DD in triangles PABPAB and PDCPDC meet at KK and the bisectors of ∠PAB\angle PAB and ∠PDC\angle PDC meet at LL. Prove that KL⊥BCKL\perp BC.
geometrycyclic quadrilateral
Two sequences

Source: Iran MO 3rd round 2019 mid-terms - Number theory P1

8/1/2019
Given a number k∈Nk\in \mathbb{N}. {an}n≥0\{a_{n}\}_{n\geq 0} and {bn}n≥0\{b_{n}\}_{n\geq 0} are two sequences of positive integers that ai,bi∈{1,2,⋯ ,9}a_{i},b_{i}\in \{1,2,\cdots,9\}. For all n≥0n\geq 0 an⋯a1a0‾+k | bn⋯b1b0‾+k.\left.\overline{a_{n}\cdots a_{1}a_{0}}+k \ \middle| \ \overline{b_{n}\cdots b_{1}b_{0}}+k \right. . Prove that there is a number 1≤t≤91\leq t \leq 9 and N∈NN\in \mathbb{N} such that bn=tanb_n=ta_n for all n≥Nn\geq N.\\ (Note that (xnxn−1…x0‾)=10n×xn+⋯+10×x1+x0(\overline{x_nx_{n-1}\dots x_0}) = 10^n\times x_n + \dots + 10\times x_1 + x_0)
number theory
Iran geometry

Source: Iran MO 3rd round 2019 finals - Geometry P1

8/14/2019
Consider a triangle ABCABC with incenter II. Let DD be the intersection of BI,ACBI,AC and CICI intersects the circumcircle of ABCABC at MM. Point KK lies on the line MDMD and ∠KIA=90∘\angle KIA=90^\circ. Let FF be the reflection of BB about CC. Prove that BIKFBIKF is cyclic.
geometryincentercircumcirclegeometric transformationreflectionLaw of SinesMenelaus
Complex geo in two consecutive years in Iran

Source: Iranian third round 2019 finals Algebra exam problem 1

8/27/2019
Let A1,A2,…AkA_1,A_2, \dots A_k be points on the unit circle.Prove that:
∑1≤i<j≤kd(Ai,Aj)2≤k2\sum\limits_{1\le i<j \le k} d(A_i,A_j)^2 \le k^2
Where d(Ai,Aj)d(A_i,A_j) denotes the distance between Ai,AjA_i,A_j.
geometryalgebra
Splitting into pathes with minimum length

Source: Iranian third round 2019 finals Combinatorics exam problem 1

8/27/2019
A bear is in the center of the left down corner of a 100∗100100*100 square .we call a cycle in this grid a bear cycle if it visits each square exactly ones and gets back to the place it started.Removing a row or column with compose the bear cycle into number of pathes.Find the minimum kk so that in any bear cycle we can remove a row or column so that the maximum length of the remaining pathes is at most kk.
combinatorics
Number theoric function eqution

Source: Iranian third round 2019 Fianls number theory exam problem 1

8/15/2019
Find all functions f:N→Nf:\mathbb{N} \to \mathbb{N} so that for any distinct positive integers x,y,zx,y,z the value of x+y+zx+y+z is a perfect square if and only if f(x)+f(y)+f(z)f(x)+f(y)+f(z) is a perfect square.
number theory