How Big?

If the sides of the triangle in the diagram are 3, 4 and 5, what is the area of the shaded square?

Do Unto Caesar

At the beginning of the night three poker players; Alan, Bernie and Craig had money in the ratios 7 : 6 : 5. At the end of the night the ratio was 6 : 5 : 4. One of them won $1 200. What were the assets of the players at the beginning of the evening? Oh for the Mathematics of Yesteryear A garrison of 600 men has just enough bread ... but, with the news that the enemy was planning an attack... How many ounces of bread a day must each man in the garrison be allowed, to hold out 45 days against the siege of the enemy? Punky Fish Stage: 3 Short Challenge Level: Every fish has at least 8 stripes, so if we had 10 fish, there would be 80 stripes with 6 stripes not yet counted. As a male fish has one extra stripe, these 6 stripes must belong to 6 male fish. So there are 6 males and 4 females. Therefore the ratio of male to female fish is 3:2. Alternatively, if there are f females and m males then$86=8f+9m=8(f+m)+m$. Now$86=8\times 10+6=8 \times 9+14...$Only the first of these is feasible so m=6 and f=4. Therefore the ratio of male to female fish is$3:2\$.

This problem is taken from the UKMT Mathematical Challenges.
{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "url": "http://nrich.maths.org/5755/solution", "fetch_time": 1505951776000000000, "content_mime_type": "text/html", "warc_filename": "crawl-data/CC-MAIN-2017-39/segments/1505818687582.7/warc/CC-MAIN-20170920232245-20170921012245-00195.warc.gz", "warc_record_offset": 260088586, "warc_record_length": 4964, "token_count": 348, "char_count": 1251, "score": 3.859375, "int_score": 4, "crawl": "CC-MAIN-2017-39", "snapshot_type": "latest", "language": "en", "language_score": 0.9511426091194153, "file_path": "/sailhome/yjruan/project/latent-lm/data/processed/finemath-4plus/finemath-4plus/train-00015-of-00064.parquet"}