2023 IOQM/Problem 1
Contents
[hide]Problem
Let be a positive integer such that . Let be the number of integers in the set
. Let , and .
Find .
Solution 1(Spacing of squares)
If for any integer , if is an integer this means is a perfect square. Now the problem reduces to finding the difference between maximum and minimum no. of perfect squares in the numbers: There are 1000 numbers here.
The idea is that for the same range of numbers, the no. of perfect squares becomes rarer when the numbers become larger.
For example, there are 3 perfect squares between 1 and 10 but none between 50 and 60.
Of course we will prove this,
Claim: The distance between 2 consecutive perfect squares gets larger as they get bigger
Proof: Let the 2 consecutive perfect squares be and . Now distance between them (Number of numbers between them) is - -1 =. So, we notice as gets larger(i.e. the perfect squares get larger as and get larger as m does), (distance between and ) also does, which means that the distance between the consecutive squares get larger as the consecutive squares get larger (As is the measure of distance between them). Hence, this proves our claim.
Now, if the distance between 2 perfect squares increases as they get larger, this suffices to prove that perfect squares get rarer as no.s get larger.
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51 52 53 54 55 56 57 58 59 60
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101 102 103 104 105 106 107 108 109 110
111 112 113 114 115 116 117 118 119 120
121 122 123 124 125 126 127 128 129 130
131 132 133 134 135 136 137 138 139 140
141 142 143 144 145 146 147 148 149 150
151 152 153 154 155 156 157 158 159 160
161 162 163 164 165 166 167 168 169 170
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661 662 663 664 665 666 667 668 669 670
671 672 673 674 675 676 677 678 679 680
681 682 683 684 685 686 687 688 689 690
691 692 693 694 695 696 697 698 699 700
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721 722 723 724 725 726 727 728 729 730
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771 772 773 774 775 776 777 778 779 780
781 782 783 784 785 786 787 788 789 790
791 792 793 794 795 796 797 798 799 800
801 802 803 804 805 806 807 808 809 810
811 812 813 814 815 816 817 818 819 820
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831 832 833 834 835 836 837 838 839 840
841 842 843 844 845 846 847 848 849 850
851 852 853 854 855 856 857 858 859 860
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871 872 873 874 875 876 877 878 879 880
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911 912 913 914 915 916 917 918 919 920
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941 942 943 944 945 946 947 948 949 950
951 952 953 954 955 956 957 958 959 960
961 962 963 964 965 966 967 968 969 970
971 972 973 974 975 976 977 978 979 980
981 982 983 984 985 986 987 988 989 990
991 992 993 994 995 996 997 998 999 1000
From this list where perfect squares are highlighted RED (Except 1 which is Blue), it's evident that perfect squares get rarer as numbers get bigger and bigger.
⇒ The maximum value of occurs when is minimum and the minimum value of occurs when is maximum.
Minimum value of = 1 So, the numbers are 5, 6...1004. there are 29 perfect squares here, so
= ()=
Maximum value of = 1000 So, the numbers are 4001, 4002...5000. there are 7 perfect squares here, so = ()=
⇒
~SANSGANKRSNGUPTA
Solution 2
All of the sets from will have 1000 elements the most the number of square numbers will be in and least number of perfect squares in . So and . This is because the gaps between square number is less when n is greater then when n is less. By checking for there would be squares from {}, a total of 29 numbers while in there would be squares from {} a total of 7 numbers, so and , giving us
~ Lakshya Pamecha
Video Solutions
Video solution by cheetna: https://www.youtube.com/watch?v=kfEyX5yBdJo
Video solution by Unacademy Olympiad Corner: https://www.youtube.com/watch?v=Mm6mXjwU9bY
Video solution by Vedantu Olympiad School: https://www.youtube.com/watch?v=4DJXtR4VHEA
Video solution by Olympiad Wallah: https://www.youtube.com/watch?v=4HSjmY7d3nA
Video solution by : Motion Olympiad Foundation Class 5th - 10th: https://www.youtube.com/watch?v=oVaeHceHXsQ
Please note that above videos solutions are in Hindi, some in English and some in mixed(Hindi + English).
~SANSGANKRSNGUPTA
See Also
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