JavaScript (ES6),  52 51  50 bytes

Saved 1 byte thanks to @tsh

Takes input as a string. Returns a Boolean value.

f=n=>n>[n%10]?f(-(-n[0]-n)%10+n.slice(1,-1)):!n[1]

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Commented

f = n =>                 // f = recursive function taking n (a string)

  n > [n % 10]           // The last digit is isolated with n % 10 and turned into a

                         // singleton array, which is eventually coerced to a string

                         // when the comparison occurs.

                         // So we do a lexicographical comparison between n and its

                         // last digit (e.g. '231'>'1' and '202'>'2', but '213'<'3').

  ?                      // If the above result is true:

    f(                   //   do a recursive call:

      -(-n[0] - n) % 10  //     We compute (int(first_digit) + int(n)) mod 10. There's no

                         //     need to isolate the last digit since we do a mod 10 anyway.

      + n.slice(1, -1)   //     We add the middle part, as a string. It may be empty.

    )                    //   end of recursive call

  :                      // else:

    !n[1]                //   return true if n has only 1 digit, or false otherwise

Perl 6, 63 62 bytes

{!grep {.[*-1]>.[0]},(.comb,{.[0,*-1].sum%10,|.[1..*-2]}...1)}

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Explanation:

{                                                            } # Anonymous code block

                     (                                  ... )       # Create a sequence

                      .comb,  # Starting with the input converted to a list of digits

                            {                          }   # With each element being

                             .[0,*-1]   # The first and last element of the previous list

                                     .sum%10  # Summed and modulo 10

                                            ,|.[1..*-2]   # Followed by the intermediate elements

                                                        ...1 # Until the list is length 1

 !grep   # Do none of the elements of the sequence

       {.[*-1]>.[0]},   # Have the last element larger than the first?

Java (JDK), 83 bytes

n->{int r=0,h=n;while(h>9)h/=10;for(;n>9;h=(h+n)%10,n/=10)r=h<n%10?1:r;return r<1;}

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Credits

• -1 byte thanks to Kevin Cruijssen
  

Jelly, 11 bytes

Ṛṙ-µṖÄ%⁵:ḊẠ

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How it works

Ṛṙ-µṖÄ%⁵:ḊẠ  Main link. Argument: n



Ṛ            Reverse n, after casting it to a digit list.

 ṙ-          Rotate the result -1 units to the left, i.e., 1 unit to the right.

             Let's call the resulting digit list D.

   µ         Begin a new chain with argument D.

    Ṗ        Pop; remove the last digit.

     Ä       Accumulate; take the cumulative sum of the remaining digits.

      %⁵     Take the sums modulo 10.

         Ḋ   Dequeue; yield D without its first digit.

        :    Perform integer division between the results to both sides.

             Integer division is truthy iff greater-or-equal is truthy.

          Ạ  All; return 1 if all quotients are truthy, 0 if not.

Mathematica, 62 bytes

0(IntegerDigits@#//.{a_,r___,b_}/;a>=b:>{Mod[a+b,10],r})=={0}&

First calls IntegerDigits on the input to get a list of its digits,
then repeatedly applies the following rule:

{a_,r___,b_}       (* match the first digit, 0 or more medial digits, and the last digit... *)

/;a>=b             (* under the condition that the number is edible... *)

:>{Mod[a+b,10],r}  (* and replace it with the next iteration *)

The rule is applied until the pattern no longer matches,
in which case either there is only one digit left (truthy)
or the head is less than the tail (falsy).

Instead of calling Length[__]==1,
we can save a few bytes with 0(__)=={0},
multiplying all elements in the list by 0
and then comparing with the list {0}.

Python 3, 50 bytes

First line stolen from Black Owl Kai's answer.

p,*s=map(int,input())

while s:*s,k=s;p%10<k>q;p+=k

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Output is via exit code. Fails (1) for falsy inputs and finishes (0) for truthy inputs.

Python 2, 105 82 81 bytes

i,x=map(int,input()),1

for y in i[:0:-1]:

 if i[0]%10<y:x=0

 else:i[0]+=y

print x

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Many thanks for a massive -23 from @ØrjanJohansen

Thanks to @VedantKandoi (and @ØrjanJohansen) for another -1

APL (Dyalog Unicode), 33 bytes^SBCS

Anonymous tacit prefix function taking a string as argument.

{⊃⍵<t←⊃⌽⍵:0⋄3::1⋄∇10|t+@1⊢¯1↓⍵}⍎¨

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⍎¨ evaluate each character (this gives us a list of digits)

{…} apply the following "dfn" to that; ⍵ is the argument (list of digits):

  ⌽⍵ reverse the argument

  ⊃ pick the first element (this is the tail)

  t← assign to t (for tail)

  ⍵< for each of the original digits, see if it is less than that

  ⊃ pick the first true/false

 : if so:

  0 return false

 ⋄ then:

 3:: if from now on, an index error (out of bounds) happens:

  1 return true

  ¯1↓⍵ drop the last digit

  ⊢ yield that (separates 1 and ¯1 so they won't form a single array)

  t+@1 add the tail to the first digit (the head)

  10| mod-10

  ∇ recurse

Once we hit a single digit, ¯1↓ will make that an empty list, and @1 will cause an index error as there is no first digit, causing the function to return true.

Python 3, 77 bytes

p,*s=map(int,input())

print(all(n<=sum(s[i+1:],p)%10for i,n in enumerate(s)))

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And my old solution with a recursive approach

Python 3, 90 bytes

f=lambda x,a=0:2>len(x)if 2>len(x)or(int(x[0])+a)%10<int(x[-1])else f(x[:-1],a+int(x[-1]))

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Takes input as a string.

Brachylog, 23 bytes

ẹ{bkK&⟨h{≥₁+tg}t⟩,K↰|Ȯ}

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This is a 1 byte save over Fatalize's solution.
This uses a recursive approach instead of a iterative

Explanation

ẹ                          Input into a list of digits

 {                    }    Assert that either

  bk                       | the list has at least 2 elements (see later)

      ⟨h{     }t⟩           | and that [head, tail]

         ≥₁                |  | is increasing (head >= tail)

           +               |  | and their sum

            t              |  | mod 10 (take last digit)

             g             |  | as single element of a list

                ,          | concatenated with

  bkK            K         | the number without head and tail (this constrains the number to have at least 2 digits)

                  ↰        | and that this constraint also works on the newly formed number

                   |Ȯ      | OR is a 1 digit number

Brachylog, 24 bytes

ẹ;I⟨⟨h{≥₁+tg}t⟩c{bk}⟩ⁱ⁾Ȯ

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I should change ⁱ's default behaviour so that it iterates an unknown number of times (currently, it iterates 1 time by default which is completely useless). I then wouldn't need the […];I[…]⁾, saving 3 bytes

Explanation

This program contains an ugly fork inside a fork. There is also some plumbing needed to work on lists of digits instead of numbers (because if we remove the head and tail of 76 we are left with 0, which doesn't work contrary to [7,6] where we end up with ).

ẹ                          Input into a list of digits

                       Ȯ   While we don't have a single digit number…

 ;I                  ⁱ⁾    …Iterate I times (I being unknown)…

   ⟨                ⟩      …The following fork:

               c           | Concatenate…

    ⟨         ⟩            | The output of the following fork:

     h       t             | | Take the head H and tail T of the number

      {     }              | | Then apply on [H,T]:

       ≥₁                  | | | H ≥ T

         +                 | | | Sum them

          tg               | | | Take the last digit (i.e. mod 10) and wrap it in a list

                {  }       | With the output of the following predicate:

                 bk        | | Remove the head and the tail of the number

Perl 5, 64 bytes

{local$_=pop;1while s/(.)(.*)(.)/$1<$3?'':(($1+$3)%10).$2/e;/./}

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Python 2, 75 67 bytes

l=lambda n:n==n[0]or n[-1]<=n[0]*l(`int(n[0]+n[-1],11)%10`+n[1:-1])

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Recursive lambda approach. Takes input as a string. Many thanks to Dennis for saving 8 bytes!

Haskell, 69 64 bytes

f n=read[show n!!0]#n

h#n=n<10||h>=mod n 10&&mod(h+n)10#div n 10

Try it online! Example usage: f 2632 yields True.

Edit: -5 bytes because mod (h + mod n 10) 10 = mod (h + n) 10

Haskell, 70 64 60 bytes

f(a:b)=b==""||a>=last b&&f(last(show$read[a]+read b):init b)

Input is taken as a string.

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Edit: -6 bytes by using @Laikoni's trick of using || instead of separate guards. Another -4 bytes thanks to @Laikoni.

Retina 0.8.2, 42 bytes

d

$*#;

^((#*).*;)2;$

$2$1

}`#{10}



^#*;$

Try it online! Link include test cases. Explanation:

d

$*#;

Convert the digits to unary and insert separators.

^((#*).*;)2;$

$2$1

If the last digit is not greater than the first then add them together.

#{10}

Reduce modulo 10 if appropriate.

}`

Repeat until the last digit is greater than the first or there is only one digit left.

^#*;$

Test whether there is only one digit left.

05AB1E, 26 25 24 bytes

[DgD#ÁD2£D`›i0qëSOθs¦¦«

Can probably be golfed a bit more.. It feels overly long, but maybe the challenge is in terms of code more complex than I thought beforehand.

Try it online or verify all test cases.

Explanation:

[                 # Start an infinite loop

 D                #  Duplicate the top of the stack

                  #  (which is the input implicitly in the first iteration)

  gD              #  Take its length and duplicate it

    #             #  If its a single digit:

                  #   Stop the infinite loop

                  #   (and output the duplicated length of 1 (truthy) implicitly)

                  #  Else:

                 #   Discard the duplicate length

   Á              #   Rotate the digits once towards the left

    D2£           #   Duplicate, and take the first two digits of the rotated number

       D`         #   Duplicate, and push the digits loose to the stack

         ›i       #   If the first digit is larger than the second digit:

           0      #    Push a 0 (falsey)

            q     #    Stop the program (and output 0 implicitly)

          ë       #   Else (first digit is smaller than or equal to the second digit):

           SO     #    Sum the two digits

             θ    #    Leave only the last digit of that sum

           s      #    Swap to take the rotated number

            ¦¦    #    Remove the first two digits

              «   #    Merge it together with the calculated new digit

C++ (gcc), 144 bytes

bool f(std::string b){int c=b.length();while(c>1){if(b[0]<b[c-1])return 0;else{b[0]=(b[0]-48+b[c-1]-48)%10+48;b=b.substr(0,c-1);--c;}}return 1;}

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First time I'm trying something like this so if I formatted something wrong please let me know. I'm not 100% sure on the rules for stuff like using namespace to eliminate the 5 bytes "std::" so I left it in.

Ungolfed:

bool hunger(std::string input)

{

    int count=input.length();

    while (count>1)

    {

        if (input[0]<input[count-1])                         //if at any point the head cannot eat the tail we can quit the loop

                                                             //comparisons can be done without switching from ascii

        {

            return false;

        }

        else

        {

             input[0]=(input[0]-48+input[count-1]-48)%10+48; //eating operation has to occur on integers so subtract 48 to convert from ASCII to a number

             input=input.substr(0,count-1);                  //get rid of the last number since it was eaten

             --count;

        }



    }

    return true;                                             //if the end of the loop is reached the number has eaten itself

}

C#, 114 bytes

static bool L(string n){return n.Skip(1).Reverse().Select(x=>x%16).Aggregate(n[0]%16,(h,x)=>h>=x?(h+x)%10:-1)>=0;}

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C (gcc) (with string.h), 110 108 bytes

c;f(char*b){c=strlen(b);if(!c)return 1;char*d=b+c-1;if(*b<*d)return 0;*b=(*b+*d-96)%10+48;*d=0;return f(b);}

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Still relatively new to PPCG, so the correct syntax for linking libraries as a new language is foreign to me. Also note that the function returns 0 or 1 for false/true, and printing that result to stdout does require stdio. If we're being pedantic and the exercise requires output, the language requires stdio that as well.

Conceptually similar to @BenH's answer, but in C, so kudos where they are due (Welcome to PPCG, btw), but using recursion. It also uses array pointer arithmetic, because dirty code is shorter than clean code.

The function is tail recursive, with exit conditions if the first number can't eat the last or the length is 1, returning false or true, respectively. These values are found by dereferencing a pointer to the C-String (which gives a char) at the beginning and end of the string, and doing the comparisons on them. pointer arithmetic is done to find the end of the string. finally, the last char is "erased" by replacing it with a null terminator (0).

It's possible that the modulus arithmetic could be shortened by a byte or two, but I already need a shower after that pointer manipulation.

Ungolfed Version Here

Update: Saved two bytes by replacing c==1 with !c. This is essentially c == 0. It will run an additional time, and will needlessly double itself before deleting itself, but saves two bytes. A side effect is null or zero length strings won't cause infinite recursion (though we shouldn't get null strings as the exercise says positive integers).

Powershell, 89 bytes

"$args"-notmatch'(.)(.*)(.)'-or(($m=$Matches).1-ge$m.3-and(.g(''+(+$m.1+$m.3)%10+$m.2)))

Important! The script calls itself recursively. So save the script as g.ps1 file in the current directory. Also you can call a script block variable instead script file (see the test script below). That calls has same length.

Note 1: The script uses a lazy evaluation of logic operators -or and -and. If "$args"-notmatch'(.)(.*)(.)' is True then the right subexpression of -or is not evaluated. Also if ($m=$Matches).1-ge$m.3 is False then the right subexpression of -and is not evaluated too. So we avoid infinite recursion.

Note 2: The regular expression '(.)(.*)(.)' does not contain start and end anchors because the expression (.*) is greedy by default.

Test script

$g={

"$args"-notmatch'(.)(.*)(.)'-or(($m=$Matches).1-ge$m.3-and(&$g(''+(+$m.1+$m.3)%10+$m.2)))

}



@(

    ,(2632, $true)

    ,(92258, $true)

    ,(60282, $true)

    ,(38410, $true)

    ,(3210, $true)

    ,(2302, $true)

    ,(2742, $true)

    ,(8628, $true)

    ,(6793, $true)

    ,(1, $true)

    ,(2, $true)

    ,(10, $true)

    ,(100, $true)

    ,(55, $true)

    ,(121, $true)

    ,(6724, $false)

    ,(47, $false)

    ,(472, $false)

    ,(60247, $false)

    ,(33265, $false)

    ,(79350, $false)

    ,(83147, $false)

    ,(93101, $false)

    ,(57088, $false)

    ,(69513, $false)

    ,(62738, $false)

    ,(54754, $false)

    ,(23931, $false)

    ,(7164, $false)

    ,(5289, $false)

    ,(3435, $false)

    ,(3949, $false)

    ,(8630, $false)

    ,(5018, $false)

    ,(6715, $false)

    ,(340, $false)

    ,(2194, $false)

) | %{

    $n,$expected = $_

   #$result = .g $n   # uncomment this line to call a script file g.ps1

    $result = &$g $n   # uncomment this line to call a script block variable $g

                       # the script block call and the script file call has same length

    "$($result-eq-$expected): $result <- $n"

}

Output:

True: True <- 2632

True: True <- 92258

True: True <- 60282

True: True <- 38410

True: True <- 3210

True: True <- 2302

True: True <- 2742

True: True <- 8628

True: True <- 6793

True: True <- 1

True: True <- 2

True: True <- 10

True: True <- 100

True: True <- 55

True: True <- 121

True: False <- 6724

True: False <- 47

True: False <- 472

True: False <- 60247

True: False <- 33265

True: False <- 79350

True: False <- 83147

True: False <- 93101

True: False <- 57088

True: False <- 69513

True: False <- 62738

True: False <- 54754

True: False <- 23931

True: False <- 7164

True: False <- 5289

True: False <- 3435

True: False <- 3949

True: False <- 8630

True: False <- 5018

True: False <- 6715

True: False <- 340

True: False <- 2194

Powershell, 90 bytes

No recursion. No file name dependency and no script block name dependency.

for($s="$args";$s[1]-and$s-ge$s%10){$s=''+(2+$s[0]+$s)%10+($s|% S*g 1($s.Length-2))}!$s[1]

A Powershell implicitly converts a right operand to a type of a left operand. Therefore, $s-ge$s%10 calculates right operand $s%10 as integer and compare it as a string because type of the left operand is string. And 2+$s[0]+$s converts a char $s[0] and string $s to integer because left operand 2 is integer.

$s|% S*g 1($s.Length-2) is a shortcut to $s.Substring(1,($s.Length-2))

C# (Visual C# Interactive Compiler), 69 bytes

x=>{for(int h=x[0],i=x.Length;i>1;)h=h<x[--i]?h/0:48+(h+x[i]-96)%10;}

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Success or failure is determined by presence or absence of an exception. Input is in the form of a string.

Less golfed...

// x is the input as a string

x=>{

  // h is the head

  for(int h=x[0],

    // i is an index to the tail

    i=x.Length;

    // continue until the tail is at 0

    i>1;)

      // is head smaller larger than tail?

      h=h<x[--i]

        // throw an exception

        ?h/0

        // calculate the next head

        :48+(h+x[i]-96)%10;

}

There are a couple extra bytes to deal with converting between characters and digits, but overall that didn't affect the size too much.

PowerShell, 94 91 bytes

for(;$n-and$n[0]-ge$n[-1]){$n=$n-replace"^.",((+$n[0]+$n[-1]-96)%10)-replace".$"}return !$n

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Test Script

function f($n){



for(;$n-and$n[0]-ge$n[-1]){$n=$n-replace"^.",((+$n[0]+$n[-1]-96)%10)-replace".$"}return !$n



Remove-Variable n

}

Write-Host True values:

@(2632, 92258, 60282, 38410,3210, 2302, 2742, 8628, 6793, 1, 2, 10, 100, 55, 121) |

    % { Write-Host "  $_ $(' '*(6-$_.ToString().Length)) $(f $_.ToString())" }



Write-Host False values:

@(6724, 47, 472, 60247, 33265, 79350, 83147, 93101, 57088, 69513, 62738, 54754, 23931, 7164, 5289, 3435, 3949, 8630, 5018, 6715, 340, 2194) | 

    % { Write-Host "  $_ $(' '*(6-$_.ToString().Length)) $(f $_.ToString())" }

Ungolfed code:

function f ($inVar){

    # While the Input exists, and the first character is greater than last

    while($inVar -and ($inVar[0] -ge $inVar[-1])){



        # Calculate the first integer -> ((+$n[0]+$n[-1]-96)%10)

        $summationChars = [int]$inVar[0]+ [int]$inVar[-1]

        # $summationChars adds the ascii values, not the integer literals. 

        $summationResult = $summationChars - 48*2

        $summationModulo = $summationResult % 10



        # Replace first character with the modulo

        $inVar = $inVar -replace "^.", $summationModulo



        # Remove last character

        $inVar = $inVar -replace ".$"

    }

    # Either it doesn't exist (Returns $True), or 

    # it exists since $inVar[0] < $inVar[-1] returning $False

    return !$inVar

}

Perl 5 -pF, 53 bytes

$F[0]=($F[0]+pop@F)%10while$#F&&$F[0]>=$F[-1];$_=!$#F

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