{"id":543,"date":"2015-04-05T04:00:27","date_gmt":"2015-04-05T03:00:27","guid":{"rendered":"http:\/\/xmau.com\/wp\/ilpost\/?p=543"},"modified":"2015-04-01T15:06:41","modified_gmt":"2015-04-01T14:06:41","slug":"problemini-per-pasqua-2015","status":"publish","type":"post","link":"https:\/\/xmau.com\/ilpost\/2015\/04\/05\/problemini-per-pasqua-2015\/","title":{"rendered":"Problemini per Pasqua 2015"},"content":{"rendered":"

Stavolta i problemini sono tratti da varie gare matematiche (per giovani, non preoccupatevi…) e recuperati dal sito Gifted Mathematics<\/a>. <\/p>\n

1. Non troppa area<\/b><\/p>\n

All’interno (quindi non sul perimetro!) di un quadrato di lato 1 ci sono nove punti. Dimostrare che se ne possono trovare tre che formano un triangolo di area minore di 1\/8. Ovviamente tre punti collineari formano un triangolo di area 0. <\/p>\n

2. Radici, solo radici<\/b><\/p>\n

Calcolate il valore (ammesso che sia finito…) dell’espressione qui sotto. <\/p>\n

\"sqrt(6+sqrt(6+sqrt(6+...)))\"<\/p>\n

3. Prodotti notevoli<\/b><\/p>\n

A parte la soluzione banale (0,0), quante altre soluzioni intere (positive o negative) ci sono per la seguente equazione?
\n(m<\/i>2<\/sup>+n<\/i>)(m<\/i>+n<\/i>2<\/sup>) = (m<\/i>+n<\/i>)3<\/sup><\/p>\n

4. Cancellazioni<\/b><\/p>\n

Vi viene detto che in una delle soluzioni dell’equazione
\nx<\/i>1<\/sub>x<\/i>2<\/sub> + x<\/i>2<\/sub>x<\/i>3<\/sub> + x<\/i>3<\/sub>x<\/i>4<\/sub> + … + x<\/i>n<\/i><\/sub>x<\/i>1<\/sub> = 0 tutti gli x<\/i>i<\/i><\/sub> hanno valore +1 oppure −1. Dimostrate che n<\/i> dev’essere multiplo di 4.<\/p>\n

5. Zigzag<\/b><\/p>\n

\"zigzag\" Il triangolo ABC<\/i> qui a fianco ha i lati lunghi 4. Il punto A1<\/sub><\/i> \u00e8 a met\u00e0 del segmento AC<\/i> e il segmento A1<\/sub>B1<\/sub><\/i> \u00e8 perpendicolare ad AC<\/i>; similmente il punto A<\/i>2<\/sub> \u00e8 a met\u00e0 del segmento A1<\/sub>C<\/i> e il segmento A<\/i>2<\/sub>B<\/i>2<\/sub> \u00e8 perpendicolare ad A<\/i>1<\/sub>C<\/i>, e cos\u00ec via all’infinito. Quanto vale la somma di tutti i segmenti B<\/i>A<\/i>1<\/sub>+B<\/i>1<\/sub>A<\/i>2<\/sub>+B<\/i>2<\/sub>A<\/i>3<\/sub>+…? <\/p>\n","protected":false},"excerpt":{"rendered":"

I soliti cinque problemini…<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center 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