MathDB

1990 Iran MO (2nd round)

Part of Iran MO (2nd Round)

Subcontests

(3)
3
2

The sum is less than 2n+1 [Iran Second Round 1990]

(a) For every positive integer nn prove that 1+122+132+⋯+1n2<21+\frac{1}{2^2}+\frac{1}{3^2}+\cdots+\frac{1}{n^2} <2
(b) Let X={1,2,3,…,n} (n≥1)X=\{1, 2, 3 ,\ldots, n\} \ ( n \geq 1) and let AkA_k be non-empty subsets of X (k=1,2,3,…,2n−1).X \ (k=1,2,3, \ldots , 2^n -1). If aka_k be the product of all elements of the set Ak,A_k, prove that ∑i=1m∑j=1m1ai⋅j2<2n+1\sum_{i=1}^{m} \sum_{j=1}^m \frac{1}{a_i \cdot j^2} <2n+1

Can we cover a 5 X 137 rectangular? [Iran Second Round 1990]

We want to cover a rectangular 5×1375 \times 137 with the following figures, prove that this is impossible.
Squars are the same and all are 1×1\text{Squars are the same and all are } \Huge{1 \times 1} [asy] import graph; size(400); real lsf = 0.5; pen dp = linewidth(0.7) + fontsize(10); defaultpen(dp); pen ds = black; pen xdxdff = rgb(0.49,0.49,1); draw((2,4)--(0,4),linewidth(2pt)); draw((0,4)--(0,0),linewidth(2pt)); draw((0,0)--(2,0),linewidth(2pt)); draw((2,0)--(2,1),linewidth(2pt)); draw((2,1)--(0,1),linewidth(2pt)); draw((1,0)--(1,4),linewidth(2pt)); draw((2,4)--(2,3),linewidth(2pt)); draw((2,3)--(0,3),linewidth(2pt)); draw((0,2)--(1,2),linewidth(2pt)); label("(1)", (0.56,-1.54), SE*lsf); draw((4,2)--(4,1),linewidth(2pt)); draw((7,2)--(7,1),linewidth(2pt)); draw((4,2)--(7,2),linewidth(2pt)); draw((4,1)--(7,1),linewidth(2pt)); draw((6,0)--(6,3),linewidth(2pt)); draw((5,3)--(5,0),linewidth(2pt)); draw((5,0)--(6,0),linewidth(2pt)); draw((5,3)--(6,3),linewidth(2pt)); label("(2)", (5.13,-1.46), SE*lsf); draw((9,0)--(9,3),linewidth(2pt)); draw((10,3)--(10,0),linewidth(2pt)); draw((12,3)--(12,0),linewidth(2pt)); draw((11,0)--(11,3),linewidth(2pt)); draw((9,2)--(12,2),linewidth(2pt)); draw((12,1)--(9,1),linewidth(2pt)); draw((9,3)--(10,3),linewidth(2pt)); draw((11,3)--(12,3),linewidth(2pt)); draw((12,0)--(11,0),linewidth(2pt)); draw((9,0)--(10,0),linewidth(2pt)); label("(3)", (10.08,-1.48), SE*lsf); draw((14,1)--(17,1),linewidth(2pt)); draw((15,2)--(17,2),linewidth(2pt)); draw((15,2)--(15,0),linewidth(2pt)); draw((15,0)--(14,0)); draw((14,1)--(14,0),linewidth(2pt)); draw((16,2)--(16,0),linewidth(2pt)); label("(4)", (15.22,-1.5), SE*lsf); draw((14,0)--(16,0),linewidth(2pt)); draw((17,2)--(17,1),linewidth(2pt)); draw((19,3)--(19,0),linewidth(2pt)); draw((20,3)--(20,0),linewidth(2pt)); draw((20,3)--(19,3),linewidth(2pt)); draw((19,2)--(20,2),linewidth(2pt)); draw((19,1)--(20,1),linewidth(2pt)); draw((20,0)--(19,0),linewidth(2pt)); label("(5)", (19.11,-1.5), SE*lsf); dot((0,0),ds); dot((0,1),ds); dot((0,2),ds); dot((0,3),ds); dot((0,4),ds); dot((1,4),ds); dot((2,4),ds); dot((2,3),ds); dot((1,3),ds); dot((1,2),ds); dot((1,1),ds); dot((2,1),ds); dot((2,0),ds); dot((1,0),ds); dot((5,0),ds); dot((6,0),ds); dot((5,1),ds); dot((6,1),ds); dot((5,2),ds); dot((6,2),ds); dot((5,3),ds); dot((6,3),ds); dot((7,2),ds); dot((7,1),ds); dot((4,1),ds); dot((4,2),ds); dot((9,0),ds); dot((9,1),ds); dot((9,2),ds); dot((9,3),ds); dot((10,0),ds); dot((11,0),ds); dot((12,0),ds); dot((10,1),ds); dot((10,2),ds); dot((10,3),ds); dot((11,1),ds); dot((11,2),ds); dot((11,3),ds); dot((12,1),ds); dot((12,2),ds); dot((12,3),ds); dot((14,0),ds); dot((15,0),ds); dot((16,0),ds); dot((15,1),ds); dot((14,1),ds); dot((16,1),ds); dot((15,2),ds); dot((16,2),ds); dot((17,2),ds); dot((17,1),ds); dot((19,0),ds); dot((20,0),ds); dot((19,1),ds); dot((20,1),ds); dot((19,2),ds); dot((20,2),ds); dot((19,3),ds); dot((20,3),ds); clip((-0.41,-10.15)--(-0.41,8.08)--(21.25,8.08)--(21.25,-10.15)--cycle); [/asy]
2
2
1
2

Maximum value of the sum [Iran Second Round 1990]

(a) Consider the set of all triangles ABCABC which are inscribed in a circle with radius R.R. When is AB2+BC2+CA2AB^2+BC^2+CA^2 maximum? Find this maximum.
(b) Consider the set of all tetragonals ABCDABCD which are inscribed in a sphere with radius R.R. When is the sum of squares of the six edges of ABCDABCD maximum? Find this maximum, and in this case prove that all of the edges are equal.

Prove that RS || Delta [Iran Second Round 1990]

Let ABCDABCD be a parallelogram. The line Δ\Delta meets the lines AB,BC,CDAB, BC, CD and DADA at M,N,PM, N, P and Q,Q, respectively. Let RR be the intersection point of the lines AB,DNAB,DN and let SS be intersection point of the lines AD,BP.AD, BP. Prove that RS∥Δ.RS \parallel \Delta.
[asy] import graph; size(400); real lsf = 0.5; pen dp = linewidth(0.7) + fontsize(10); defaultpen(dp); pen ds = black; pen xdxdff = rgb(0.49,0.49,1); pen qqzzcc = rgb(0,0.6,0.8); pen wwwwff = rgb(0.4,0.4,1); draw((2,2)--(6,2),qqzzcc+linewidth(1.6pt)); draw((6,2)--(4,0),qqzzcc+linewidth(1.6pt)); draw((-1.95,(+12-2*-1.95)/2)--(12.24,(+12-2*12.24)/2),qqzzcc+linewidth(1.6pt)); draw((-1.95,(-0+3*-1.95)/3)--(12.24,(-0+3*12.24)/3),qqzzcc+linewidth(1.6pt)); draw((-1.95,(-0-0*-1.95)/6)--(12.24,(-0-0*12.24)/6),qqzzcc+linewidth(1.6pt)); draw((4,0)--(4,4),wwwwff+linewidth(1.2pt)+linetype("3pt 3pt")); draw((2,2)--(8.14,0),wwwwff+linewidth(1.2pt)+linetype("3pt 3pt")); draw((-1.95,(+32.56-4*-1.95)/4.14)--(12.24,(+32.56-4*12.24)/4.14),qqzzcc+linewidth(1.6pt)); dot((0,0),ds); label("AA", (0,-0.3),NE*lsf); dot((4,0),ds); label("BB", (4.02,-0.33),NE*lsf); dot((2,2),ds); label("DD", (1.81,2.07),NE*lsf); dot((6,2),ds); label("CC", (6.16,2.08),NE*lsf); dot((3,3),ds); label("QQ", (2.97,3.22),NE*lsf); dot((5,1),ds); label("NN", (4.99,1.19),NE*lsf); label("Δ\Delta", (1.7,3.76),NE*lsf); dot((6,0),ds); label("MM", (5.9,-0.33),NE*lsf); dot((4,2),ds); label("PP", (4.02,2.08),NE*lsf); dot((4,4),ds); label("SS", (3.94,4.12),NE*lsf); dot((8.14,0),ds); label("EE", (8.2,0.09),NE*lsf); clip((-1.95,-6.96)--(-1.95,4.99)--(12.24,4.99)--(12.24,-6.96)--cycle); [/asy]