PAT/source/html/a1068.html

Find More Coins
<div class="ques-view"><p>Eva loves to collect coins from all over the universe, including some other planets like Mars.  One day she visited a universal shopping mall which could accept all kinds of coins as payments.  However, there was a special requirement of the payment: for each bill, she must pay the exact amount.  Since she has as many as <span class="katex"><span class="katex-mathml"><math><mrow><mn>1</mn><msup><mn>0</mn><mn>4</mn></msup></mrow>10^4</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.8141079999999999em;"></span><span class="strut bottom" style="height:0.8141079999999999em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathrm">1</span><span class="mord"><span class="mord mathrm">0</span><span class="msupsub"><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped mtight"><span class="mord mathrm mtight">4</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span></span> coins with her, she definitely needs your help.  You are supposed to tell her, for any given amount of money, whether or not she can find some coins to pay for it.</p>
<h3 id="input-specification-">Input Specification:</h3>
<p>Each input file contains one test case.  For each case, the first line contains 2 positive numbers: <code>N</code> (<span class="katex"><span class="katex-mathml"><math><mrow><mo>≤</mo><mn>1</mn><msup><mn>0</mn><mn>4</mn></msup></mrow>\le 10^4</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.8141079999999999em;"></span><span class="strut bottom" style="height:0.950078em;vertical-align:-0.13597em;"></span><span class="base textstyle uncramped"><span class="mrel">≤</span><span class="mord mathrm">1</span><span class="mord"><span class="mord mathrm">0</span><span class="msupsub"><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped mtight"><span class="mord mathrm mtight">4</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span></span>, the total number of coins) and <code>M</code> (<span class="katex"><span class="katex-mathml"><math><mrow><mo>≤</mo><mn>1</mn><msup><mn>0</mn><mn>2</mn></msup></mrow>\le 10^2</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.8141079999999999em;"></span><span class="strut bottom" style="height:0.950078em;vertical-align:-0.13597em;"></span><span class="base textstyle uncramped"><span class="mrel">≤</span><span class="mord mathrm">1</span><span class="mord"><span class="mord mathrm">0</span><span class="msupsub"><span class="vlist"><span style="top:-0.363em;margin-right:0.05em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle uncramped mtight"><span class="mord mathrm mtight">2</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span></span>, the amount of money Eva has to pay).  The second line contains <code>N</code> face values of the coins, which are all positive numbers.  All the numbers in a line are separated by a space.</p>
<h3 id="output-specification-">Output Specification:</h3>
<p>For each test case, print in one line the face values <span class="katex"><span class="katex-mathml"><math><mrow><msub><mi>V</mi><mn>1</mn></msub><mo>≤</mo><msub><mi>V</mi><mn>2</mn></msub><mo>≤</mo><mo>⋯</mo><mo>≤</mo><msub><mi>V</mi><mi>k</mi></msub></mrow>V_1 \le V_2 \le \cdots \le V_k</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.68333em;"></span><span class="strut bottom" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit" style="margin-right:0.22222em;">V</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.22222em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord mathrm mtight">1</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span><span class="mrel">≤</span><span class="mord"><span class="mord mathit" style="margin-right:0.22222em;">V</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.22222em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord mathrm mtight">2</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span><span class="mrel">≤</span><span class="minner">⋯</span><span class="mrel">≤</span><span class="mord"><span class="mord mathit" style="margin-right:0.22222em;">V</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.22222em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord mathit mtight" style="margin-right:0.03148em;">k</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span></span> such that <span class="katex"><span class="katex-mathml"><math><mrow><msub><mi>V</mi><mn>1</mn></msub><mo>+</mo><msub><mi>V</mi><mn>2</mn></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mi>V</mi><mi>k</mi></msub><mo>=</mo></mrow>V_1 + V_2 + \cdots + V_k = </math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.68333em;"></span><span class="strut bottom" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit" style="margin-right:0.22222em;">V</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.22222em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord mathrm mtight">1</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span><span class="mbin">+</span><span class="mord"><span class="mord mathit" style="margin-right:0.22222em;">V</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.22222em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord mathrm mtight">2</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span><span class="mbin">+</span><span class="minner">⋯</span><span class="mbin">+</span><span class="mord"><span class="mord mathit" style="margin-right:0.22222em;">V</span><span class="msupsub"><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.22222em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped mtight"><span class="mord mathit mtight" style="margin-right:0.03148em;">k</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span><span class="mrel">=</span></span></span></span><code>M</code>.  All the numbers must be separated by a space, and there must be no extra space at the end of the line.  If such a solution is not unique, output the smallest sequence.  If there is no solution, output "No Solution" instead.</p>
<p>Note: sequence {A[1], A[2], ...} is said to be "smaller" than sequence  {B[1], B[2], ...} if there exists <span class="katex"><span class="katex-mathml"><math><mrow><mi>k</mi><mo>≥</mo><mn>1</mn></mrow>k \ge 1</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.69444em;"></span><span class="strut bottom" style="height:0.83041em;vertical-align:-0.13597em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.03148em;">k</span><span class="mrel">≥</span><span class="mord mathrm">1</span></span></span></span> such that A[<span class="katex"><span class="katex-mathml"><math><mrow><mi>i</mi></mrow>i</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.65952em;"></span><span class="strut bottom" style="height:0.65952em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit">i</span></span></span></span>]=B[<span class="katex"><span class="katex-mathml"><math><mrow><mi>i</mi></mrow>i</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.65952em;"></span><span class="strut bottom" style="height:0.65952em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit">i</span></span></span></span>] for all <span class="katex"><span class="katex-mathml"><math><mrow><mi>i</mi><mo>&lt;</mo><mi>k</mi></mrow>i &lt; k</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.69444em;"></span><span class="strut bottom" style="height:0.73354em;vertical-align:-0.0391em;"></span><span class="base textstyle uncramped"><span class="mord mathit">i</span><span class="mrel">&lt;</span><span class="mord mathit" style="margin-right:0.03148em;">k</span></span></span></span>, and A[<span class="katex"><span class="katex-mathml"><math><mrow><mi>k</mi></mrow>k</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.69444em;"></span><span class="strut bottom" style="height:0.69444em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.03148em;">k</span></span></span></span>] <span class="katex"><span class="katex-mathml"><math><mrow><mo>&lt;</mo></mrow>&lt;</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.5391em;"></span><span class="strut bottom" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="base textstyle uncramped"><span class="mrel">&lt;</span></span></span></span> B[<span class="katex"><span class="katex-mathml"><math><mrow><mi>k</mi></mrow>k</math></span><span aria-hidden="true" class="katex-html"><span class="strut" style="height:0.69444em;"></span><span class="strut bottom" style="height:0.69444em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.03148em;">k</span></span></span></span>].</p>
<h3 id="sample-input-1-">Sample Input 1:</h3>
<pre><code class="lang-in">8 9
5 9 8 7 2 3 4 1
</code></pre>
<h3 id="sample-output-1-">Sample Output 1:</h3>
<pre><code class="lang-out">1 3 5
</code></pre>
<h3 id="sample-input-2-">Sample Input 2:</h3>
<pre><code>4 8
7 2 4 3
</code></pre><h3 id="sample-output-2-">Sample Output 2:</h3>
<pre><code>No Solution
</code></pre></div>