Understand the problem

Show that for each non-negative integer  there are unique non-negative integers  and  such that we have

Start with hints

Observe that  \( \frac{(x+y)^2 + 3x + y}{2} = \frac{(x+y)^2 + (x+y)}{2} + x\) is the expression what we get after bringing in the symmetry.Now, factorize it and see what we are looking for is \( n = \frac{(x+y)(x+y+1)}{2} + x\). Can you guess anything about the expression \( \frac{(x+y)(x+y+1)}{2} \).

\( \frac{(k)(k+1)}{2} \)  is the sum of the first k natural numbers. So, now the idea is that somehow you are taking the first k natural numbers and adding another number x to it to make any number.Can you get the final logic?  [/et_pb_tab][et_pb_tab title="Hint 4" _builder_version="4.0" hover_enabled="0"]Now, observe that sum of the first k numbers is increasing with k. Now, take any number say 17. Now observe that 17 lies between 15 and 21. This means that 17 = 15 + 2 = ( 1 +2 + 3 +4 + 5) +  2.So, this is the idea that k = x+y is the largest number such that n is greater than or equal to the ( 1 + 2 + ... + k ) and observe that we may need something like 2 in case of n = 17.Call that x and obviously \( k \geq x \). Hence, define y = k - x.So, for n = 33 = ( 1 + 2 + ... +  7 ) + 5. Hence k = x + y = 7, x = 5, y = 2. This is the whole idea.QED.  [/et_pb_tab][/et_pb_tabs][et_pb_text _builder_version="3.27.4" text_font="Raleway|300|||||||" text_text_color="#ffffff" header_font="Raleway|300|||||||" header_text_color="#e2e2e2" background_color="#0c71c3" custom_margin="48px||48px" custom_padding="20px|20px|20px|20px" border_radii="on|5px|5px|5px|5px" box_shadow_style="preset3"]

Watch video

Connected Program at Cheenta

Math Olympiad is the greatest and most challenging academic contest for school students. Brilliant school students from over 100 countries participate in it every year.Cheenta works with small groups of gifted students through an intense training program. It is a deeply personalized journey toward intellectual prowess and technical sophistication.[/et_pb_blurb][et_pb_button button_url="https://cheenta.com/matholympiad/" url_new_window="on" button_text="Learn More" button_alignment="center" _builder_version="3.23.3" custom_button="on" button_bg_color="#0c71c3" button_border_color="#0c71c3" button_border_radius="0px" button_font="Raleway||||||||" button_icon="%%3%%" background_layout="dark" button_text_shadow_style="preset1" box_shadow_style="preset1" box_shadow_color="#0c71c3"][/et_pb_button][et_pb_text _builder_version="3.27.4" text_font="Raleway|300|||||||" text_text_color="#ffffff" header_font="Raleway|300|||||||" header_text_color="#e2e2e2" background_color="#0c71c3" custom_margin="50px||50px" custom_padding="20px|20px|20px|20px" border_radii="on|5px|5px|5px|5px" box_shadow_style="preset3"]

Similar Problems