Marinada '24 - Školjka

Vrijeme: 09:06

Konstruktivne kritike | Constructive criticism #4

Neka je $\overline{OG}$ dužina duljine 4 i neka je $\sigma$ kružnica sa središtem u $O$ i radijusom 2. Neka je točka $X$ takva da je $\angle XOG=90^\circ$ i $|OX|=1.$ Neka su $l_1$ i $l_2$ redom unutarnja i vanjska simetrala kuta $\angle OXG.$ Neka $l_1$ i $l_2$ sijeku pravac $OG$ u $Z$ i $W$ redom. Kružnica sa središtem u $Z$ i polumjerom $\overline{ZX}$ siječe pravac $OG$ u $F$ i $A,$ tako da je $|GA|<|GF|.$ Kružnica sa središtem u $W$ i polumjerom $\overline{WX}$ siječe pravac $OG$ u $H$ i $B,$ tako da je $|GB|<|GH|.$ Neka je $T$ takva da je $|AT|=|OB|,$ $AT \perp OG,$ te su $T$ i $X$ s iste strane pravca $OG.$ Neka kružnica promjera $\overline{TX}$ siječe pravac $OG$ u točkama $D$ i $P,$ tako da je $|GP|<|GD|.$ Neka okomica na $OG$ u $P$ siječe $\sigma$ u točkama $M$ i $N.$ Odredi kut $\angle MOG.$

Budući da znamo da ste prethodna 3 zadatka mogli trivijalno nacrati i izmjeriti u geogebri, u želji da izbjegnemo takvo ponašanje i ovdje, ovog puta vas tražimo sljedeće: traženi $\angle MOG$ želimo zapisati u stupnjevima, u decimalnom zapisu (dakle ne želimo minute i sekunde, nego decimalni broj, npr. želimo $34.5^\circ$ a ne $34^\circ 30'$ ) i želimo da kao odgovor napišete znamenke od 81. do 96. (uključivo) decimalnog mjesta (počinjemo brojati nakon decimalne točke, dakle u broju 76.542, znamenka 5 je na prvom mjestu, 4 je na drugom mjestu i 2 je na trećem mjestu). Dakle, vaš odgovor mora biti tih 16 znamenki, zapisane kao jedan 16-teroznamenkasti broj.

Let $\overline{OG}$ be a segment of length 4, and let $\sigma$ be a circle centered at $O$ with a radius of 2. Let point $X$ be such that $\angle XOG=90^\circ$ and $|OX|=1$ . Let $l_1$ and $l_2$ be the internal and external angle bisectors of angle $\angle OXG$ , respectively. Let $l_1$ and $l_2$ intersect line $OG$ at points $Z$ and $W$ , respectively. The circle centered at $Z$ with radius $\overline{ZX}$ intersects line $OG$ at points $F$ and $A$ , such that $|GA|<|GF|$ . The circle centered at $W$ with radius $\overline{WX}$ intersects line $OG$ at points $H$ and $B$ , such that $|GB|<|GH|$ . Let $T$ be such that $|AT|=|OB|$ , $AT \perp OG$ , and points $T$ and $X$ are on the same side of line $OG$ . The circle with diameter $\overline{TX}$ intersects line $OG$ at points $D$ and $P$ , such that $|GP|<|GD|$ . A perpendicular to $OG$ at $P$ intersects $\sigma$ at points $M$ and $N$ . Determine the angle $\angle MOG$ .

Since we know that you could easily draw and measure the previous three problems using GeoGebra, in order to avoid such behavior here, this time we ask you the following: we want the requested angle $\angle MOG$ to be expressed in degrees, in decimal form (meaning we do not want minutes and seconds, but a decimal number, e.g., we want $34.5^\circ$ and not $34^\circ 30'$ ). We also want you to provide the digits from the 81st to the 96th (inclusive) decimal place (starting to count after the decimal point, so in the number 76.542, the digit 5 is in the first position, 4 is in the second position, and 2 is in the third position). Therefore, your answer must consist of those 16 digits, written as a single 16-digit number.

[lang=hr] Neka je $\overline{OG}$ dužina duljine 4 i neka je $\sigma$ kružnica sa središtem u $O$ i radijusom 2. Neka je točka $X$ takva da je $\angle XOG=90^\circ$ i $|OX|=1.$ Neka su $l_1$ i $l_2$ redom unutarnja i vanjska simetrala kuta $\angle OXG.$ Neka $l_1$ i $l_2$ sijeku pravac $OG$ u $Z$ i $W$ redom. Kružnica sa središtem u $Z$ i polumjerom $\overline{ZX}$ siječe pravac $OG$ u $F$ i $A,$ tako da je $|GA|<|GF|.$ Kružnica sa središtem u $W$ i polumjerom $\overline{WX}$ siječe pravac $OG$ u $H$ i $B,$ tako da je $|GB|<|GH|.$ Neka je $T$ takva da je $|AT|=|OB|,$ $AT \perp OG,$ te su $T$ i $X$ s iste strane pravca $OG.$ Neka kružnica promjera $\overline{TX}$ siječe pravac $OG$ u točkama $D$ i $P,$ tako da je $|GP|<|GD|.$ Neka okomica na $OG$ u $P$ siječe $\sigma$ u točkama $M$ i $N.$ Odredi kut $\angle MOG.$ Budući da znamo da ste prethodna 3 zadatka mogli trivijalno nacrati i izmjeriti u geogebri, u želji da izbjegnemo takvo ponašanje i ovdje, ovog puta vas tražimo sljedeće: traženi $\angle MOG$ želimo zapisati u stupnjevima, u decimalnom zapisu (dakle ne želimo minute i sekunde, nego decimalni broj, npr. želimo $34.5^\circ$ a ne $34^\circ 30'$) i želimo da kao odgovor napišete znamenke od 81. do 96. (uključivo) decimalnog mjesta (počinjemo brojati nakon decimalne točke, dakle u broju 76.542, znamenka 5 je na prvom mjestu, 4 je na drugom mjestu i 2 je na trećem mjestu). Dakle, vaš odgovor mora biti tih 16 znamenki, zapisane kao jedan 16-teroznamenkasti broj. [/lang] [lang=en] Let $\overline{OG}$ be a segment of length 4, and let $\sigma$ be a circle centered at $O$ with a radius of 2. Let point $X$ be such that $\angle XOG=90^\circ$ and $|OX|=1$. Let $l_1$ and $l_2$ be the internal and external angle bisectors of angle $\angle OXG$, respectively. Let $l_1$ and $l_2$ intersect line $OG$ at points $Z$ and $W$, respectively. The circle centered at $Z$ with radius $\overline{ZX}$ intersects line $OG$ at points $F$ and $A$, such that $|GA|<|GF|$. The circle centered at $W$ with radius $\overline{WX}$ intersects line $OG$ at points $H$ and $B$, such that $|GB|<|GH|$. Let $T$ be such that $|AT|=|OB|$, $AT \perp OG$, and points $T$ and $X$ are on the same side of line $OG$. The circle with diameter $\overline{TX}$ intersects line $OG$ at points $D$ and $P$, such that $|GP|<|GD|$. A perpendicular to $OG$ at $P$ intersects $\sigma$ at points $M$ and $N$. Determine the angle $\angle MOG$. Since we know that you could easily draw and measure the previous three problems using GeoGebra, in order to avoid such behavior here, this time we ask you the following: we want the requested angle $\angle MOG$ to be expressed in degrees, in decimal form (meaning we do not want minutes and seconds, but a decimal number, e.g., we want $34.5^\circ$ and not $34^\circ 30'$). We also want you to provide the digits from the 81st to the 96th (inclusive) decimal place (starting to count after the decimal point, so in the number 76.542, the digit 5 is in the first position, 4 is in the second position, and 2 is in the third position). Therefore, your answer must consist of those 16 digits, written as a single 16-digit number. [/lang]