Datasets:

ai4bharat
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indic-frontier-math

id large_stringlengths 48 71	tier int64 1 4	title large_stringlengths 29 52	problem large_stringlengths 100 1.68k	answer large_stringlengths 3 74	solution large_stringlengths 761 20k	source_url large_stringclasses 1 value	problem_Assamese_translation large_stringlengths 98 1.85k	problem_Bengali_translation large_stringlengths 99 1.85k	problem_Gujarati_translation large_stringlengths 90 1.96k	problem_Hindi_translation large_stringlengths 86 1.88k	problem_Kannada_translation large_stringlengths 102 1.87k	problem_Malayalam_translation large_stringlengths 133 2.01k	problem_Marathi_translation large_stringlengths 106 1.96k	problem_Nepali_translation large_stringlengths 95 1.84k	problem_Odia_translation large_stringlengths 99 1.78k	problem_Punjabi_translation large_stringlengths 92 1.86k	problem_Sanskrit_translation large_stringlengths 110 1.88k	problem_Tamil_translation large_stringlengths 120 1.87k	problem_Telugu_translation large_stringlengths 98 1.86k	problem_Urdu_translation large_stringlengths 107 1.85k
frontiermath-tier4-an-optimization-problem-in-bmo-space	4	An optimization problem in BMO space	Let \[ c = \sup_{f} \int_{0}^{1} (f(t)^{3}+\|f(t)\|)dt\] where supremum is taken over all integrable functions $f : [0,1] \to \mathbb{R}$ such that $\int_{0}^{1}f(t)dt=-10$, $\int_{0}^{1} f^{2}(t)dt=100+\frac{1}{12}$, and \[ \sup_{J \subset [0,1]} \frac{1}{\|J\|}\int_{J} \left\|f(t) - \frac{1}{\|J\|}\int_{J}f(s)ds \...	$\frac{\sqrt{3}}{36} + \frac{\sqrt{3}}{6} e^{-20\sqrt{3}-\frac{1}{6}}$	Let us consider the following optimization problem \[\begin{align*} &B(x_{1}, x_{2}; \varepsilon) \stackrel{\mathrm{def}}{=}\\ &\sup_{f \in L^{1}([0,1])}\left\{\int_{0}^{1}F(f(t))dt\, : \, \int_{0}^{1} f(t)dt=x_{1}, \, \int_{0}^{1}f^{2}(t) dt=x_{2},\, \sup_{J \subset [0,1]} \frac{1}{\|J\|}\int_{J}\left\|f(t) - \frac{...	https://epoch.ai/frontiermath/tiers-1-4/benchmark-problems	ধৰা হওক \[ c = \sup_{f} \int_{0}^{1} (f(t)^{3}+\|f(t)\|)dt\] য’ত supremum লোৱা হৈছে এনে সকলো অনুকলনযোগ্য ফলন $f : [0,1] \to \mathbb{R}$-ৰ ওপৰত যাতে $\int_{0}^{1}f(t)dt=-10$, $\int_{0}^{1} f^{2}(t)dt=100+\frac{1}{12}$, আৰু \[ \sup_{J \subset [0,1]} \frac{1}{\|J\|}\int_{J} \left\|f(t) - \frac{1}{\|J\|}\int_{J}f(s)ds \...	ধরি, \[ c = \sup_{f} \int_{0}^{1} (f(t)^{3}+\|f(t)\|)dt\] যেখানে সুপ্রিমামটি এমন সকল সমাকলনযোগ্য ফাংশন $f : [0,1] \to \mathbb{R}$-এর উপর নেওয়া হয়েছে যেন $\int_{0}^{1}f(t)dt=-10$, $\int_{0}^{1} f^{2}(t)dt=100+\frac{1}{12}$, এবং \[ \sup_{J \subset [0,1]} \frac{1}{\|J\|}\int_{J} \left\|f(t) - \frac{1}{\|J\|}\int_{J}f...	ધારો કે \[ c = \sup_{f} \int_{0}^{1} (f(t)^{3}+\|f(t)\|)dt\] જ્યાં સુપ્રિમમ એવા તમામ સંકલનીય વિધેયો $f : [0,1] \to \mathbb{R}$ પર લેવામાં આવે છે કે જેથી $\int_{0}^{1}f(t)dt=-10$, $\int_{0}^{1} f^{2}(t)dt=100+\frac{1}{12}$, અને \[ \sup_{J \subset [0,1]} \frac{1}{\|J\|}\int_{J} \left\|f(t) - \frac{1}{\|J\|}\int_{J}f(s...	मान लीजिए कि \[ c = \sup_{f} \int_{0}^{1} (f(t)^{3}+\|f(t)\|)dt\] जहाँ supremum ऐसे सभी समाकलनीय फलनों $f : [0,1] \to \mathbb{R}$ पर लिया गया है जिनके लिए $\int_{0}^{1}f(t)dt=-10$, $\int_{0}^{1} f^{2}(t)dt=100+\frac{1}{12}$, और \[ \sup_{J \subset [0,1]} \frac{1}{\|J\|}\int_{J} \left\|f(t) - \frac{1}{\|J\|}\int_{J}f(...	\[ c = \sup_{f} \int_{0}^{1} (f(t)^{3}+\|f(t)\|)dt\] ಇರಲಿ, ಇಲ್ಲಿ ಸುಪ್ರಿಮಮ್ ಅನ್ನು $\int_{0}^{1}f(t)dt=-10$, $\int_{0}^{1} f^{2}(t)dt=100+\frac{1}{12}$, ಮತ್ತು \[ \sup_{J \subset [0,1]} \frac{1}{\|J\|}\int_{J} \left\|f(t) - \frac{1}{\|J\|}\int_{J}f(s)ds \right\|^{2}dt \leq \frac{1}{12},\] ಆಗಿರುವಂತಹ ಎಲ್ಲಾ ಸಮಾಕಲನೀಯ ಫಲನಗಳು \(...	\[ c = \sup_{f} \int_{0}^{1} (f(t)^{3}+\|f(t)\|)dt\] ആയിരിക്കട്ടെ, ഇവിടെ സുപ്രീമം എന്നത് $\int_{0}^{1}f(t)dt=-10$, $\int_{0}^{1} f^{2}(t)dt=100+\frac{1}{12}$, കൂടാതെ \[ \sup_{J \subset [0,1]} \frac{1}{\|J\|}\int_{J} \left\|f(t) - \frac{1}{\|J\|}\int_{J}f(s)ds \right\|^{2}dt \leq \frac{1}{12},\] എന്നിങ്ങനെയുള്ള എല്ലാ ഇന്...	समजा \[ c = \sup_{f} \int_{0}^{1} (f(t)^{3}+\|f(t)\|)dt\] येथे supremum अशा सर्व संकलनीय (integrable) फलनांवर $f : [0,1] \to \mathbb{R}$ घेतला आहे ज्यांच्यासाठी $\int_{0}^{1}f(t)dt=-10$, $\int_{0}^{1} f^{2}(t)dt=100+\frac{1}{12}$, आणि \[ \sup_{J \subset [0,1]} \frac{1}{\|J\|}\int_{J} \left\|f(t) - \frac{1}{\|J\|}\in...	मानौँ \[ c = \sup_{f} \int_{0}^{1} (f(t)^{3}+\|f(t)\|)dt\] जहाँ supremum $\int_{0}^{1}f(t)dt=-10$, $\int_{0}^{1} f^{2}(t)dt=100+\frac{1}{12}$, र \[ \sup_{J \subset [0,1]} \frac{1}{\|J\|}\int_{J} \left\|f(t) - \frac{1}{\|J\|}\int_{J}f(s)ds \right\|^{2}dt \leq \frac{1}{12},\] हुने गरी सबै समाकलनीय (integrable) फलनहरू \(f...	ଧରାଯାଉ \[ c = \sup_{f} \int_{0}^{1} (f(t)^{3}+\|f(t)\|)dt\] ଯେଉଁଠାରେ ସୁପ୍ରିମମ୍ (supremum) ସେହି ସମସ୍ତ ସମାକଳନୀୟ ଫଳନ $f : [0,1] \to \mathbb{R}$ ଗୁଡ଼ିକ ଉପରେ ନିଆଯାଇଛି ଯେପରିକି $\int_{0}^{1}f(t)dt=-10$, $\int_{0}^{1} f^{2}(t)dt=100+\frac{1}{12}$, ଏବଂ \[ \sup_{J \subset [0,1]} \frac{1}{\|J\|}\int_{J} \left\|f(t) - \frac{1...	ਮੰਨ ਲਓ \[ c = \sup_{f} \int_{0}^{1} (f(t)^{3}+\|f(t)\|)dt\] ਜਿੱਥੇ supremum ਉਹਨਾਂ ਸਾਰੇ integrable ਫੰਕਸ਼ਨਾਂ $f : [0,1] \to \mathbb{R}$ ਉੱਤੇ ਲਿਆ ਗਿਆ ਹੈ ਤਾਂ ਜੋ $\int_{0}^{1}f(t)dt=-10$, $\int_{0}^{1} f^{2}(t)dt=100+\frac{1}{12}$, ਅਤੇ \[ \sup_{J \subset [0,1]} \frac{1}{\|J\|}\int_{J} \left\|f(t) - \frac{1}{\|J\|}\int_{J}...	अस्तु \[ c = \sup_{f} \int_{0}^{1} (f(t)^{3}+\|f(t)\|)dt\] यत्र उच्चिष्ठः (supremum) तादृशानां सर्वेषां समाकलनीय-फलनानां (integrable functions) $f : [0,1] \to \mathbb{R}$ उपरि गृहीतः यैः $\int_{0}^{1}f(t)dt=-10$, $\int_{0}^{1} f^{2}(t)dt=100+\frac{1}{12}$, तथा \[ \sup_{J \subset [0,1]} \frac{1}{\|J\|}\int_{J} \le...	\[ c = \sup_{f} \int_{0}^{1} (f(t)^{3}+\|f(t)\|)dt\] என்க. இங்கு $\int_{0}^{1}f(t)dt=-10$, $\int_{0}^{1} f^{2}(t)dt=100+\frac{1}{12}$, மற்றும் \[ \sup_{J \subset [0,1]} \frac{1}{\|J\|}\int_{J} \left\|f(t) - \frac{1}{\|J\|}\int_{J}f(s)ds \right\|^{2}dt \leq \frac{1}{12},\] என்பவற்றை நிறைவு செய்யும் அனைத்து தொகையிடத்தக்க ...	\[ c = \sup_{f} \int_{0}^{1} (f(t)^{3}+\|f(t)\|)dt\] అనుకుందాం, ఇక్కడ సుప్రిమమ్ అనేది $\int_{0}^{1}f(t)dt=-10$, $\int_{0}^{1} f^{2}(t)dt=100+\frac{1}{12}$, మరియు \[ \sup_{J \subset [0,1]} \frac{1}{\|J\|}\int_{J} \left\|f(t) - \frac{1}{\|J\|}\int_{J}f(s)ds \right\|^{2}dt \leq \frac{1}{12},\] అయ్యేటట్లుగా ఉండే అన్ని సమాకల...	فرض کریں کہ \[ c = \sup_{f} \int_{0}^{1} (f(t)^{3}+\|f(t)\|)dt\] جہاں supremum ان تمام قابلِ تکامل تفاعلات $f : [0,1] \to \mathbb{R}$ پر لیا گیا ہے جن کے لیے $\int_{0}^{1}f(t)dt=-10$، $\int_{0}^{1} f^{2}(t)dt=100+\frac{1}{12}$، اور \[ \sup_{J \subset [0,1]} \frac{1}{\|J\|}\int_{J} \left\|f(t) - \frac{1}{\|J\|}\int_{...
frontiermath-tier4-elliptic-curves-and-higher-galois-theory	4	Elliptic curves and higher Galois Theory	Define functions $F(z)$ and $G(z)$ on the upper-half of the complex plane by $F(z):=1+\sum_{n=1}^{\infty} \frac{e^{2\pi i n^2z}}{\prod_{j=1}^n (1+e^{2 \pi i jz})^2} \ \ \ \textrm{and}$ $G(z):=\prod_{n=1}^{\infty} \frac{(1-e^{2\pi i nz})(1-e^{2\pi i (2n-1)z})}{(1+e^{2\pi i n z})}.$ Let $\ell_1$ be the smalles...	73948492097301691765464030714938921180352979774791349212169829996615700482	We first determine the pair $(\ell_1, \ell_2)$ by examining conditions (1)-(4). Conditions (1) and (2) are standard in algebraic number theory. Condition (1) demands (see Ch. 5.1 of [1]) that $D_{\ell_1}:=-\ell_1$ is the discriminant of the ring of integers of the imaginary quadratic field \(\mathbb Q(\sqrt{-\e...	https://epoch.ai/frontiermath/tiers-1-4/benchmark-problems	জটিল সমতলৰ উচ্চ-অৰ্ধাংশত ফলন $F(z)$ আৰু $G(z)$ ক তলত দিয়া ধৰণে সংজ্ঞায়িত কৰক $F(z):=1+\sum_{n=1}^{\infty} \frac{e^{2\pi i n^2z}}{\prod_{j=1}^n (1+e^{2 \pi i jz})^2} \ \ \ \textrm{আৰু}$ $G(z):=\prod_{n=1}^{\infty} \frac{(1-e^{2\pi i nz})(1-e^{2\pi i (2n-1)z})}{(1+e^{2\pi i n z})}.$ ধৰক $\ell_1$ হৈছে এনে আটাইত...	জটিল তলের ঊর্ধ্ব-অর্ধে (upper-half of the complex plane) $F(z)$ এবং $G(z)$ ফাংশন দুটি নিম্নোক্তভাবে সংজ্ঞায়িত করুন: $F(z):=1+\sum_{n=1}^{\infty} \frac{e^{2\pi i n^2z}}{\prod_{j=1}^n (1+e^{2 \pi i jz})^2} \ \ \ \textrm{এবং}$ \(G(z):=\prod_{n=1}^{\infty} \frac{(1-e^{2\pi i nz})(1-e^{2\pi i (2n-1)z})}{(1+e^{2\pi i ...	સંકર સમતલના ઉપલા અર્ધભાગ (upper-half of the complex plane) પર વિધેયો $F(z)$ અને $G(z)$ ને નીચે પ્રમાણે વ્યાખ્યાયિત કરો: $F(z):=1+\sum_{n=1}^{\infty} \frac{e^{2\pi i n^2z}}{\prod_{j=1}^n (1+e^{2 \pi i jz})^2} \ \ \ \textrm{અને}$ \(G(z):=\prod_{n=1}^{\infty} \frac{(1-e^{2\pi i nz})(1-e^{2\pi i (2n-1)z})}{(1+e^{2\pi...	सम्मिश्र तल के ऊपरी अर्ध-तल पर फलनों $F(z)$ और $G(z)$ को निम्न प्रकार परिभाषित कीजिए: $F(z):=1+\sum_{n=1}^{\infty} \frac{e^{2\pi i n^2z}}{\prod_{j=1}^n (1+e^{2 \pi i jz})^2} \ \ \ \textrm{तथा}$ $G(z):=\prod_{n=1}^{\infty} \frac{(1-e^{2\pi i nz})(1-e^{2\pi i (2n-1)z})}{(1+e^{2\pi i n z})}.$ मान लीजिए $\ell_1$...	ಸಂಕೀರ್ಣ ಸಮತಲದ ಮೇಲಿನ ಅರ್ಧಭಾಗದಲ್ಲಿ $F(z)$ ಮತ್ತು $G(z)$ ಉತ್ಪನ್ನಗಳನ್ನು ಈ ಕೆಳಗಿನಂತೆ ವ್ಯಾಖ್ಯಾನಿಸಿ: $F(z):=1+\sum_{n=1}^{\infty} \frac{e^{2\pi i n^2z}}{\prod_{j=1}^n (1+e^{2 \pi i jz})^2} \ \ \ \textrm{and}$ $G(z):=\prod_{n=1}^{\infty} \frac{(1-e^{2\pi i nz})(1-e^{2\pi i (2n-1)z})}{(1+e^{2\pi i n z})}.$ ಕೆಳಗಿನ ಎಲ್ಲ ಷ...	കോംപ്ലക്സ് പ്രതലത്തിന്റെ മുകൾ പകുതിയിൽ (upper-half of the complex plane) $F(z)$, $G(z)$ എന്നീ ഫംഗ്ഷനുകൾ താഴെ പറയുന്ന വിധം നിർവചിക്കുക: $F(z):=1+\sum_{n=1}^{\infty} \frac{e^{2\pi i n^2z}}{\prod_{j=1}^n (1+e^{2 \pi i jz})^2} \ \ \ \textrm{and}$ \(G(z):=\prod_{n=1}^{\infty} \frac{(1-e^{2\pi i nz})(1-e^{2\pi i (2n-1)...	सम्मिश्र प्रतलाच्या वरील अर्धभागावर (upper-half of the complex plane) $F(z)$ आणि $G(z)$ ही फलने पुढीलप्रमाणे परिभाषित करा: $F(z):=1+\sum_{n=1}^{\infty} \frac{e^{2\pi i n^2z}}{\prod_{j=1}^n (1+e^{2 \pi i jz})^2} \ \ \ \textrm{आणि}$ \(G(z):=\prod_{n=1}^{\infty} \frac{(1-e^{2\pi i nz})(1-e^{2\pi i (2n-1)z})}{(1+e^{2...	सम्मिश्र समतलको माथिल्लो आधा भागमा फलनहरू $F(z)$ र $G(z)$ लाई निम्नद्वारा परिभाषित गर्नुहोस्: $F(z):=1+\sum_{n=1}^{\infty} \frac{e^{2\pi i n^2z}}{\prod_{j=1}^n (1+e^{2 \pi i jz})^2} \ \ \ \textrm{र}$ $G(z):=\prod_{n=1}^{\infty} \frac{(1-e^{2\pi i nz})(1-e^{2\pi i (2n-1)z})}{(1+e^{2\pi i n z})}.$ मानौं \(\ell_1...	ଜଟିଳ ସମତଳର ଉପର-ଅର୍ଦ୍ଧାଂଶ (upper-half of the complex plane) ରେ ଫଳନ $F(z)$ ଏବଂ $G(z)$ କୁ ନିମ୍ନଲିଖିତ ଭାବରେ ସଂଜ୍ଞାୟିତ କରନ୍ତୁ: $F(z):=1+\sum_{n=1}^{\infty} \frac{e^{2\pi i n^2z}}{\prod_{j=1}^n (1+e^{2 \pi i jz})^2} \ \ \ \textrm{ଏବଂ}$ \(G(z):=\prod_{n=1}^{\infty} \frac{(1-e^{2\pi i nz})(1-e^{2\pi i (2n-1)z})}{(1+e^{2\...	ਕੰਪਲੈਕਸ ਤਲ ਦੇ ਉੱਪਰਲੇ ਅੱਧ ਉੱਤੇ ਫ਼ੰਕਸ਼ਨਾਂ $F(z)$ ਅਤੇ $G(z)$ ਨੂੰ ਇਸ ਪ੍ਰਕਾਰ ਪਰਿਭਾਸ਼ਿਤ ਕਰੋ $F(z):=1+\sum_{n=1}^{\infty} \frac{e^{2\pi i n^2z}}{\prod_{j=1}^n (1+e^{2 \pi i jz})^2} \ \ \ \textrm{ਅਤੇ}$ $G(z):=\prod_{n=1}^{\infty} \frac{(1-e^{2\pi i nz})(1-e^{2\pi i (2n-1)z})}{(1+e^{2\pi i n z})}.$ ਮੰਨ ਲਓ $\ell_1$ ਉਹ...	सम्मिश्रतलस्य उपरि-अर्धभागे फलने $F(z)$ तथा $G(z)$ अनयोः रूपेण परिभाषयत: $F(z):=1+\sum_{n=1}^{\infty} \frac{e^{2\pi i n^2z}}{\prod_{j=1}^n (1+e^{2 \pi i jz})^2} \ \ \ \textrm{तथा}$ $G(z):=\prod_{n=1}^{\infty} \frac{(1-e^{2\pi i nz})(1-e^{2\pi i (2n-1)z})}{(1+e^{2\pi i n z})}.$ मान्यताम् $\ell_1$ सा लघुतमा अभ...	சிக்கலெண் தளத்தின் மேல் அரைத்தளத்தில் $F(z)$ மற்றும் $G(z)$ ஆகிய சார்புகளைப் பின்வருமாறு வரையறுக்கவும்: $F(z):=1+\sum_{n=1}^{\infty} \frac{e^{2\pi i n^2z}}{\prod_{j=1}^n (1+e^{2 \pi i jz})^2} \ \ \ \textrm{மற்றும்}$ $G(z):=\prod_{n=1}^{\infty} \frac{(1-e^{2\pi i nz})(1-e^{2\pi i (2n-1)z})}{(1+e^{2\pi i n z})}.$...	సంకీర్ణ తలం యొక్క ఎగువ అర్ధభాగంపై ప్రమేయాలు $F(z)$ మరియు $G(z)$ లను ఈ క్రింది విధంగా నిర్వచించండి: $F(z):=1+\sum_{n=1}^{\infty} \frac{e^{2\pi i n^2z}}{\prod_{j=1}^n (1+e^{2 \pi i jz})^2} \ \ \ \textrm{మరియు}$ $G(z):=\prod_{n=1}^{\infty} \frac{(1-e^{2\pi i nz})(1-e^{2\pi i (2n-1)z})}{(1+e^{2\pi i n z})}.$ క్రిం...	کمپلیکس پلین کے بالائی نصف پر تفاعلات $F(z)$ اور $G(z)$ کی تعریف یوں کریں کہ $F(z):=1+\sum_{n=1}^{\infty} \frac{e^{2\pi i n^2z}}{\prod_{j=1}^n (1+e^{2 \pi i jz})^2} \ \ \ \textrm{اور}$ $G(z):=\prod_{n=1}^{\infty} \frac{(1-e^{2\pi i nz})(1-e^{2\pi i (2n-1)z})}{(1+e^{2\pi i n z})}.$ فرض کریں کہ $\ell_1$ وہ سب ...
frontiermath-tier3-combinatorics-in-the-tsirelson-space	3	Combinatorics in the Tsirelson space	Background. The Tsirelson space is a Banach space introduced in 1974 as a counterexample to a longstanding conjecture in Banach space theory. It is a subspace of $\ell^\infty$ described as follows. Let $P_k$ denote the operator on $\ell^\infty$ that modifies each \(\langle x_n:\, n \in \mathbb N \rangle \in \ell...	402653211	Let us begin by introducing some helpful shorthand. In the first half of this explanation, we will deal with quite a few sequences of the form $\langle 0,0,\dots,0,0,b,b-\varepsilon,b-2\varepsilon,\dots,a+2\varepsilon,a+\varepsilon,a,0,0,0,\dots \rangle$ where $0 < a < b < 1$. For convenience, let \(\langle a,b,(m,...	https://epoch.ai/frontiermath/tiers-1-4/benchmark-problems	পৃষ্ঠভূমি। Tsirelson space হৈছে এটা Banach space যাক Banach space তত্ত্বৰ এক দীৰ্ঘদিনীয়া অনুমানৰ এটা counterexample হিচাপে 1974 চনত প্ৰৱৰ্তন কৰা হৈছিল। ই হৈছে $\ell^\infty$ ৰ এটা subspace যাক তলত দিয়া ধৰণে বৰ্ণনা কৰা হৈছে। ধৰা হওক $P_k$ এ $\ell^\infty$ ৰ ওপৰত এনে এক operator ক বুজায় যিয়ে প্ৰতিটো \(\langle x_...	প্রেক্ষাপট। Tsirelson স্পেস হলো একটি Banach স্পেস যা Banach স্পেস তত্ত্বের একটি দীর্ঘস্থায়ী অনুমানের পাল্টা উদাহরণ (counterexample) হিসেবে 1974 সালে প্রবর্তিত হয়েছিল। এটি $\ell^\infty$-এর একটি সাবস্পেস যা নিম্নরূপ বর্ণিত। ধরি $P_k$ হলো $\ell^\infty$-এর ওপর এমন একটি অপারেটর যা প্রতিটি \(\langle x_n:\, n \in \ma...	પૃષ્ઠભૂમિ. સાયરેલસન અવકાશ (Tsirelson space) એ 1974 માં બનાખ અવકાશ સિદ્ધાંતમાં લાંબા સમયથી ચાલતી એક કલ્પનાના પ્રત્યુદાહરણ તરીકે રજૂ કરાયેલ એક બનાખ અવકાશ છે. તે $\ell^\infty$ નો એક ઉપઅવકાશ છે જે નીચે મુજબ વર્ણવાયેલ છે. ધારો કે $P_k$ એ $\ell^\infty$ પરનો એવો પ્રક્રિયક દર્શાવે છે જે દરેક \(\langle x_n:\, n \in \math...	पृष्ठभूमि। त्सिरेलसन समष्टि (Tsirelson space) एक बानाख समष्टि (Banach space) है जिसे बानाख समष्टि सिद्धांत में एक दीर्घकालिक अनुमान (conjecture) के प्रत्युदाहरण (counterexample) के रूप में 1974 में प्रस्तुत किया गया था। यह $\ell^\infty$ की एक उपसमष्टि है जिसे निम्नानुसार वर्णित किया गया है। मान लीजिए कि $P_k$, \(\...	ಹಿನ್ನೆಲೆ. ಸಿರೆಲ್ಸನ್ ಸ್ಪೇಸ್ (Tsirelson space) ಎಂಬುದು ಬನಾಚ್ ಸ್ಪೇಸ್ ಸಿದ್ಧಾಂತದಲ್ಲಿನ ದೀರ್ಘಕಾಲದ ಊಹೆಗೆ (conjecture) ಪ್ರತಿದೃಷ್ಟಾಂತವಾಗಿ 1974 ರಲ್ಲಿ ಪರಿಚಯಿಸಲಾದ ಬನಾಚ್ ಸ್ಪೇಸ್ ಆಗಿದೆ. ಇದು ಕೆಳಗಿನಂತೆ ವಿವರಿಸಲಾದ $\ell^\infty$ ನ ಒಂದು ಉಪಸ್ಪೇಸ್ ಆಗಿದೆ. $P_k$ ಎಂಬುದು $\ell^\infty$ ಮೇಲಿನ ಆಪರೇಟರ್ ಆಗಿರಲಿ, ಇದು ಪ್ರತಿಯೊಂದು \(\langle x_n:\, n ...	പശ്ചാത്തലം. ബനാക് സ്പേസ് തിയറിയിലെ (Banach space theory) ഒരു ചിരകാല അനുമാനത്തിന് ഒരു കൗണ്ടർ എക്സാമ്പിൾ (counterexample) ആയി 1974-ൽ അവതരിപ്പിക്കപ്പെട്ട ഒരു ബനാക് സ്പേസ് ആണ് സിറൽസൺ സ്പേസ് (Tsirelson space). ഇത് താഴെ വിവരിച്ചിരിക്കുന്ന പ്രകാരം $\ell^\infty$-ന്റെ ഒരു സബ്സ്പേസ് ആണ്. $\ell^\infty$-ലെ ഓരോ \(\langle x_n:\...	पार्श्वभूमी. त्सिरेलसन अवकाश (Tsirelson space) हे बनाख अवकाश सिद्धांतातील (Banach space theory) एका प्रदीर्घ काळापासून असलेल्या अनुमानाला प्रतिउदाहरण म्हणून 1974 मध्ये मांडलेले एक बनाख अवकाश (Banach space) आहे. हे खालीलप्रमाणे वर्णन केलेले $\ell^\infty$ चे एक उपअवकाश (subspace) आहे. समजा $P_k$ हे $\ell^\infty$ व...	पृष्ठभूमि। Tsirelson space एक Banach space हो जुन सन् 1974 मा Banach space सिद्धान्तको एक लामो समयदेखिको अनुमानको प्रतिउदाहरण (counterexample) को रूपमा प्रस्तुत गरिएको थियो। यो $\ell^\infty$ को एक subspace हो जसलाई निम्नानुसार वर्णन गरिएको छ। मानौँ $P_k$ ले $\ell^\infty$ मा रहेको यस्तो operator लाई जनाउँछ जसले प...	ପୃଷ୍ଠଭୂମି: Tsirelson space ହେଉଛି Banach space theory ର ଏକ ଦୀର୍ଘସ୍ଥାୟୀ ଅନୁମାନର ଏକ ପ୍ରତି-ଉଦାହରଣ (counterexample) ଭାବରେ 1974 ମସିହାରେ ପ୍ରବର୍ତ୍ତିତ ଏକ Banach space । ଏହା ନିମ୍ନରେ ବର୍ଣ୍ଣିତ $\ell^\infty$ ର ଏକ ଉପସ୍ପେସ୍ (subspace) ଅଟେ । ମନେକର $P_k$ ହେଉଛି $\ell^\infty$ ଉପରେ ଏକ ଅପରେଟର୍ ଯାହା ପ୍ରଥମ $k$ ଟି ସ୍ଥାନାଙ୍କକୁ $0$ ସ...	ਪਿਛੋਕੜ। ਸਿਰੇਲਸਨ ਸਪੇਸ (Tsirelson space) ਇੱਕ ਬਾਨਾਕ ਸਪੇਸ (Banach space) ਹੈ ਜੋ 1974 ਵਿੱਚ ਬਾਨਾਕ ਸਪੇਸ ਸਿਧਾਂਤ ਵਿੱਚ ਇੱਕ ਲੰਬੇ ਸਮੇਂ ਤੋਂ ਚੱਲੇ ਆ ਰਹੇ ਅਨੁਮਾਨ ਦੇ ਖੰਡਨ-ਉਦਾਹਰਨ (counterexample) ਵਜੋਂ ਪੇਸ਼ ਕੀਤਾ ਗਿਆ ਸੀ। ਇਹ $\ell^\infty$ ਦਾ ਇੱਕ ਸਬਸਪੇਸ ਹੈ ਜਿਸਦਾ ਵੇਰਵਾ ਹੇਠਾਂ ਦਿੱਤਾ ਗਿਆ ਹੈ। ਮੰਨ ਲਓ $P_k$ $\ell^\infty$ ਉੱਤੇ ਉਸ ਓਪਰੇਟਰ ਨੂੰ ਦਰ...	पृष्ठभूमिः। त्सिरेल्सन-समष्टिः (Tsirelson space) काचित् बानाख-समष्टिः अस्ति, या बानाख-समष्टि-सिद्धान्ते एकस्य चिरकालिकस्यानुमानस्य प्रत्युदाहरणरूपेण 1974 तमे वर्षे प्रवर्तिता। इयम् अधोलिखितरूपेण वर्णिता $\ell^\infty$ इत्यस्य उपसमष्टिः अस्ति। मानयतु यत् $P_k$ इति $\ell^\infty$ इत्यस्मिन् सः संकारकः अस्ति, यः प्रत...	பின்னணி. Tsirelson வெளி என்பது Banach வெளி கோட்பாட்டில் நீண்டகாலமாக இருந்த ஒரு கணித ஊகத்திற்கு எதிர்ப்புதாரணமாக 1974-இல் அறிமுகப்படுத்தப்பட்ட ஒரு Banach வெளி ஆகும். இது பின்வருமாறு விவரிக்கப்படும் $\ell^\infty$-ன் ஓர் உள்வெளி ஆகும். $P_k$ என்பது $\ell^\infty$-இல் உள்ள ஒவ்வோர் \(\langle x_n:\, n \in \mathbb N \ra...	నేపథ్యం. సిరెల్సన్ స్పేస్ (Tsirelson space) అనేది బనాచ్ స్పేస్ సిద్ధాంతంలో చాలా కాలంగా ఉన్న ఒక ఊహనానికి (conjecture) ప్రతికూల ఉదాహరణగా 1974లో ప్రవేశపెట్టబడిన ఒక బనాచ్ స్పేస్. ఇది క్రింది విధంగా వివరించబడిన $\ell^\infty$ యొక్క ఉపస్పేస్. $\ell^\infty$ పై $P_k$ అనేది ప్రతి \(\langle x_n:\, n \in \mathbb N \rangle \...	پس منظر۔ سیریلسن اسپیس (Tsirelson space) ایک باناخ اسپیس (Banach space) ہے جو 1974 میں باناخ اسپیس تھیوری کے ایک دیرینہ مفروضے (conjecture) کی جوابی مثال (counterexample) کے طور پر متعارف کرائی گئی تھی۔ یہ $\ell^\infty$ کی ایک ذیلی فضا (subspace) ہے جسے درج ذیل کے طور پر بیان کیا جاتا ہے۔ فرض کریں کہ $P_k$، \(\ell...
frontiermath-tier3-enumerative-geometry-over-finite-fields	3	Enumerative geometry over finite fields	Let $U\subset\mathbb{P}H^0(\mathbb{P}^2_{\mathbb{Z}}, \mathcal{O}(2))$ be the space of smooth conics in $\mathbb{P}^2_{\mathbb{Z}}$, and let $Z\subset U^6$ be the closed subscheme parametrizing $6$-tuples $(C_1, \cdots, C_6)$ with $C_1$ tangent to $C_2, \cdots, C_6$. Let $\pi: Z\to U^5$ be the map induc...	866	The limit $L$ is $\sum_{m=1}^{3264}\frac{1}{m} \approx 8.668,$ giving an answer of 866. To see this, we use Harris’s result (see “Galois groups of enumerative problems,” from Duke Math J. 1979) that the map $\pi: Z\to V$ is degree $3264$ (known classically) with Galois group the full symmetric group \(S_{3264}\...	https://epoch.ai/frontiermath/tiers-1-4/benchmark-problems	ধৰা হওক $U\subset\mathbb{P}H^0(\mathbb{P}^2_{\mathbb{Z}}, \mathcal{O}(2))$ হ'ল $\mathbb{P}^2_{\mathbb{Z}}$ -ত মসৃণ শংকুছেদসমূহৰ স্থান, আৰু ধৰা হওক $Z\subset U^6$ হ'ল এনে $6$-টাপল $(C_1, \cdots, C_6)$ পেৰামিটাৰাইজ কৰা বন্ধ উপ-স্কিম (closed subscheme) য'ত $C_1$ হ'ল $C_2, \cdots, C_6$-ৰ স্পৰ্শক। ধৰা হওক \(\p...	ধরি $U\subset\mathbb{P}H^0(\mathbb{P}^2_{\mathbb{Z}}, \mathcal{O}(2))$ হলো $\mathbb{P}^2_{\mathbb{Z}}$-এ মসৃণ কনিকের স্পেস, এবং ধরি $Z\subset U^6$ হলো এমন একটি বদ্ধ সাবস্কিম যা $C_1$ কনিকটি $C_2, \cdots, C_6$-এর সাথে স্পর্শক বিশিষ্ট $6$-টিউপল $(C_1, \cdots, C_6)$-সমূহকে প্যারামিটারাইজ করে। ধরি \(\pi: Z\t...	ધારો કે $U\subset\mathbb{P}H^0(\mathbb{P}^2_{\mathbb{Z}}, \mathcal{O}(2))$ એ $\mathbb{P}^2_{\mathbb{Z}}$ માં સ્મૂધ કોનિક્સ (smooth conics) ની સ્પેસ છે, અને ધારો કે $Z\subset U^6$ એ $C_1$ એ $C_2, \cdots, C_6$ ને સ્પર્શક હોય તેવી $6$-ટપલ્સ $(C_1, \cdots, C_6)$ ને પેરામીટરાઇઝ કરતું ક્લોઝ્ડ સબસ્કીમ (closed su...	मान लीजिए कि $U\subset\mathbb{P}H^0(\mathbb{P}^2_{\mathbb{Z}}, \mathcal{O}(2))$, $\mathbb{P}^2_{\mathbb{Z}}$ में स्मूथ शांकवों (smooth conics) की समष्टि है, और मान लीजिए कि $Z\subset U^6$ ऐसे $6$-टुपलों $(C_1, \cdots, C_6)$ को प्राचलीकृत करने वाली संवृत उप-स्कीम (closed subscheme) है जिनमें $C_1$, \(C_2, \c...	$U\subset\mathbb{P}H^0(\mathbb{P}^2_{\mathbb{Z}}, \mathcal{O}(2))$ ಎಂಬುದು $\mathbb{P}^2_{\mathbb{Z}}$ ನಲ್ಲಿನ ನುಣುಪಾದ ಶಂಕುಜಗಳ (smooth conics) ಆಕಾಶವಾಗಿರಲಿ, ಮತ್ತು $Z\subset U^6$ ಎಂಬುದು $C_2, \cdots, C_6$ ಗಳಿಗೆ $C_1$ ಸ್ಪರ್ಶಕವಾಗಿರುವ $6$-ಟಪಲ್‌ಗಳನ್ನು $(C_1, \cdots, C_6)$ ಪ್ಯಾರಾಮೀಟರೀಕರಿಸುವ ಮುಚ್ಚಿದ ಉಪಯೋಜನೆಯಾಗಿರಲಿ ...	$\mathbb{P}^2_{\mathbb{Z}}$ ലെ smooth conics-ന്റെ space $U\subset\mathbb{P}H^0(\mathbb{P}^2_{\mathbb{Z}}, \mathcal{O}(2))$ ആയിരിക്കട്ടെ, കൂടാതെ $C_1$ എന്നത് $C_2, \cdots, C_6$ എന്നിവയിലേക്ക് tangent ആയിട്ടുള്ള $6$-tuples $(C_1, \cdots, C_6)$-നെ parametrise ചെയ്യുന്ന closed subscheme $Z\subset U^6$ ആയിരിക്...	समजा $U\subset\mathbb{P}H^0(\mathbb{P}^2_{\mathbb{Z}}, \mathcal{O}(2))$ ही $\mathbb{P}^2_{\mathbb{Z}}$ मधील smooth conics ची स्पेस आहे, आणि समजा $Z\subset U^6$ ही $C_1$ हे $C_2, \cdots, C_6$ ला tangent असणाऱ्या $6$-tuples $(C_1, \cdots, C_6)$ चे प्राचलनीकरण करणारी closed subscheme आहे. समजा \(\pi: Z\to U^...	मानौं $U\subset\mathbb{P}H^0(\mathbb{P}^2_{\mathbb{Z}}, \mathcal{O}(2))$, $\mathbb{P}^2_{\mathbb{Z}}$ मा स्मूथ शंकुहरूको स्पेस हो, र मानौं $Z\subset U^6$ भन्नाले $C_1$ लाई $C_2, \cdots, C_6$ सँग स्पर्शरेखीय हुने गरी $6$-tuples $(C_1, \cdots, C_6)$ लाई परिमितीकरण गर्ने खुला उपयोजना हो। मानौं \(\pi: Z\to U^...	ମନେକର $U\subset\mathbb{P}H^0(\mathbb{P}^2_{\mathbb{Z}}, \mathcal{O}(2))$ ହେଉଛି $\mathbb{P}^2_{\mathbb{Z}}$ ରେ smooth conics ର space, ଏବଂ ମନେକର $Z\subset U^6$ ହେଉଛି $C_1$ ଟି $C_2, \cdots, C_6$ ସହିତ tangent ଥିବା $6$-tuples $(C_1, \cdots, C_6)$ କୁ parametrize କରୁଥିବା closed subscheme। ମନେକର $\pi: Z\to U^5$...	ਮੰਨ ਲਓ ਕਿ $U\subset\mathbb{P}H^0(\mathbb{P}^2_{\mathbb{Z}}, \mathcal{O}(2))$, $\mathbb{P}^2_{\mathbb{Z}}$ ਵਿੱਚ ਸਮੂਥ ਕੌਨਿਕਾਂ (smooth conics) ਦੀ ਸਪੇਸ ਹੈ, ਅਤੇ ਮੰਨ ਲਓ ਕਿ $Z\subset U^6$ ਅਜਿਹੇ $6$-ਟਪਲਜ਼ $(C_1, \cdots, C_6)$ ਨੂੰ ਪੈਰਾਮੀਟਰਾਈਜ਼ ਕਰਨ ਵਾਲੀ ਕਲੋਜ਼ਡ ਸਬ-ਸਕੀਮ (closed subscheme) ਹੈ ਜਿਨ੍ਹਾਂ ਵਿੱਚ $C_1$, \(C_2, ...	मन्यताम् $U\subset\mathbb{P}H^0(\mathbb{P}^2_{\mathbb{Z}}, \mathcal{O}(2))$ $\mathbb{P}^2_{\mathbb{Z}}$ इत्यस्मिन् मसृणशाङ्कवानाम् अवकाशः अस्ति, तथा च $Z\subset U^6$ सा संवृतोपयोजना (closed subscheme) अस्ति या तादृशान् $6$-गणान् $(C_1, \cdots, C_6)$ प्राचलीकरोति यत्र $C_1$ इत्ययं $C_2, \cdots, C_6$ इत्येत...	$U\subset\mathbb{P}H^0(\mathbb{P}^2_{\mathbb{Z}}, \mathcal{O}(2))$ என்பது $\mathbb{P}^2_{\mathbb{Z}}$ இல் உள்ள மென்மையான கூம்புவெட்டுகளின் (smooth conics) வெளி என்க, மற்றும் $Z\subset U^6$ என்பது $C_2, \cdots, C_6$ உடன் $C_1$ தொடுநிலையில் உள்ளவாறான $6$-வரிசைத் தொகுப்புகளை ($6$-tuples) \((C_1, \cdots, C_6)...	$U\subset\mathbb{P}H^0(\mathbb{P}^2_{\mathbb{Z}}, \mathcal{O}(2))$ అనేది $\mathbb{P}^2_{\mathbb{Z}}$ లోని smooth conics యొక్క ప్రదేశం (space) అనుకుందాం, మరియు $Z\subset U^6$ అనేది $C_2, \cdots, C_6$ లకు $C_1$ tangent గా ఉండే $6$-tuples $(C_1, \cdots, C_6)$ లను పారామితికరించే closed subscheme అనుకుందాం. \(...	فرض کریں کہ $U\subset\mathbb{P}H^0(\mathbb{P}^2_{\mathbb{Z}}, \mathcal{O}(2))$، $\mathbb{P}^2_{\mathbb{Z}}$ میں ہموار مخروطات (smooth conics) کی فضا ہے، اور فرض کریں کہ $Z\subset U^6$ ایک بند ذیلی اسکیم (closed subscheme) ہے جو ایسے $6$-tuples $(C_1, \cdots, C_6)$ کو پیرامیٹرائز کرتی ہے جن میں $C_1$، مخروطا...
frontiermath-tier3-testing-artin-s-primitive-root-conjecture	3	Testing Artin’s primitive root conjecture	"For a positive integer \$n\$, let \$v_p(n)\$ denote the largest integer \$v\$ such that \\(p^(...TRUNCATED)	367707	"We will break this into multiple sections.\n\nBackground. First of all, the presence of the \$x\$(...TRUNCATED)	https://epoch.ai/frontiermath/tiers-1-4/benchmark-problems	"এটা ধনাত্মক অখণ্ড সংখ্যা \$n\$-ৰ বাবে, ধৰ(...TRUNCATED)	"একটি ধনাত্মক পূর্ণসংখ্যা \$n\$-এর জন্য, (...TRUNCATED)	"ધન પૂર્ણાંક \$n\$ માટે, ધારો કે \$v_p(n)\$ એ એવ(...TRUNCATED)	"एक धनात्मक पूर्णांक \$n\$ के लिए, माना \\(v_p(...TRUNCATED)	"ಧನಾತ್ಮಕ ಪೂರ್ಣಾಂಕ \$n\$ ಗಾಗಿ, \$p^v \\mid n\$ ಆಗಿರ(...TRUNCATED)	"ഒരു പോസിറ്റീവ് പൂർണ്ണസംഖ്യ \$n\$-ന്, \\(p^v \(...TRUNCATED)	"एका धन पूर्णांक \$n\$ साठी, \$v_p(n)\$ हा असा स(...TRUNCATED)	"एक धनात्मक पूर्णाङ्क \$n\$ का लागि, मानौ(...TRUNCATED)	"ଏକ ଧନାତ୍ମକ ପୂର୍ଣ୍ଣସଂଖ୍ୟା \$n\$ ପାଇଁ, ମନ(...TRUNCATED)	"ਇੱਕ ਧਨਾਤਮਕ ਪੂਰਨ ਅੰਕ \$n\$ ਲਈ, ਮੰਨ ਲਓ \$v_p(n)\$ (...TRUNCATED)	"कस्मैचित् धनात्मकपूर्णाङ्काय \$n\$, \\(v_p((...TRUNCATED)	"ஒரு மிகை முழு எண் \$n\$-க்கு, \$p^v \\mid n\$ என்ற(...TRUNCATED)	"ఒక ధన పూర్ణాంకం \$n\$ కొరకు, \$p^v \\mid n\$ అయ్య(...TRUNCATED)	"ایک مثبت صحیح عدد \$n\$ کے لیے، فرض کریں کہ \$v_p(n)\$ اس سب (...TRUNCATED)
frontiermath-tier2-a-recursive-construction-on-large-permutations	2	A recursive construction on large permutations	"Let \$W\$ be the set of finite words with all distinct letters over the alphabet of positive inte(...TRUNCATED)	499999999972	"For \$w\\in W\$, write \$x\\prec_w y\$ to mean \$x\$ appears before \$y\$ in \$w\$.\nSupp(...TRUNCATED)	https://epoch.ai/frontiermath/tiers-1-4/benchmark-problems	"ধৰা হওক \$W\$ হৈছে ধনাত্মক অখণ্ড সংখ্যা(...TRUNCATED)	"ধরি \$W\$ হলো ধনাত্মক পূর্ণসংখ্যার বর্ণ(...TRUNCATED)	"ધારો કે \$W\$ એ ધન પૂર્ણાંકોના મૂળાક્ષર(...TRUNCATED)	"मान लीजिए कि \$W\$ धनात्मक पूर्णांकों क(...TRUNCATED)	"\$W\$ ಎಂಬುದು ಧನಾತ್ಮಕ ಪೂರ್ಣಾಂಕಗಳ ವರ್ಣಮಾ(...TRUNCATED)	"പോസിറ്റീവ് പൂർണ്ണസംഖ്യകളുടെ ആൽഫബെറ(...TRUNCATED)	"समजा \$W\$ हा धन पूर्णांकांच्या वर्णमाल(...TRUNCATED)	"मानौं \$W\$ धनात्मक पूर्णाङ्कहरूको वर्(...TRUNCATED)	"ମନେକରନ୍ତୁ \$W\$ ହେଉଛି ଧନାତ୍ମକ ପୂର୍ଣ୍ଣସ(...TRUNCATED)	"ਮੰਨ ਲਓ \$W\$ ਧਨਾਤਮਕ ਪੂਰਨ ਅੰਕਾਂ ਦੀ ਵਰਣਮਾਲ(...TRUNCATED)	"धनात्मकपूर्णाङ्कानां वर्णमालायां स(...TRUNCATED)	"நேர்ம முழு எண்களின் அகரவரிசையின் மீ(...TRUNCATED)	"ధన పూర్ణాంకాల వర్ణమాల (alphabet) పై అన్ని వ(...TRUNCATED)	"فرض کریں کہ \$W\$ مثبت صحیح اعداد کے حروف تہجی پر تمام مم(...TRUNCATED)
frontiermath-tier2-asymptotics-of-solutions-of-a-5-variable-equation	2	Asymptotics of solutions of a 5-variable equation	"There are unique constants \$C, \\alpha, \\beta\$ such that the number of solutions to the equati(...TRUNCATED)	214	"This count is equal to\n\$ \\sum_{n\\le x}d_{2}(n)d_{3}(n+1)\$\nwhere \\(d_{k}(n) =\\displaystyle(...TRUNCATED)	https://epoch.ai/frontiermath/tiers-1-4/benchmark-problems	"এনে অনন্য ধ্ৰুৱক \$C, \\alpha, \\beta\$ আছে যাতে \\(a, (...TRUNCATED)	"এমন অনন্য ধ্রুবক \$C, \\alpha, \\beta\$ রয়েছে যেন \\((...TRUNCATED)	"એવા અનન્ય અચળાંકો \$C, \\alpha, \\beta\$ છે કે જેથી(...TRUNCATED)	"ऐसे अद्वितीय अचर \$C, \\alpha, \\beta\$ विद्यमान ह(...TRUNCATED)	"\$a, b, c, d, e \\in \\mathbb{N}\$ ಮತ್ತು \$ab \\le x\$ ಆಗಿರುವಾಗ\n\n\\(...TRUNCATED)	"\$ab \\le x\$ ആയ \$a, b, c, d, e \\in \\mathbb{N}\$ സഹിതമുള്ള\n\n\\(ab+1=(...TRUNCATED)	"\$a, b, c, d, e \\in \\mathbb{N}\$ आणि \$ab \\le x\$ सह\n\n\$ab+1=cde\$\n\nया (...TRUNCATED)	"त्यस्ता अद्वितीय अचरहरू \$C, \\alpha, \\beta\$ छन(...TRUNCATED)	"ଏପରି ଅନନ୍ୟ ଧ୍ରୁବକ \$C, \\alpha, \\beta\$ ରହିଛନ୍ତି (...TRUNCATED)	"ਅਜਿਹੇ ਵਿਲੱਖਣ ਅਚੱਲ \$C, \\alpha, \\beta\$ ਮੌਜੂਦ ਹਨ ਕ(...TRUNCATED)	"एतादृशाः अनन्याः स्थिराङ्काः \\(C, \\alpha, \\b(...TRUNCATED)	"\$x\\rightarrow\\infty\$ ஆகும்போது, \$ab \\le x\$ என்றவாறுள(...TRUNCATED)	"\$a, b, c, d, e \\in \\mathbb{N}\$ మరియు \$ab \\le x\$ తో\n\n\$ab+1=cde\$\n\n(...TRUNCATED)	"ایسے منفرد مستقلات \$C, \\alpha, \\beta\$ موجود ہیں کہ مساوات\n\n(...TRUNCATED)
frontiermath-tier2-find-the-degree-19-polynomial	2	Find the degree 19 polynomial	"Construct a degree 19 polynomial \$p(x) \\in \\mathbb{C}[x]\$ such that \\(X:=\\{p(x) = p(y)\\} \(...TRUNCATED)	1876572071974094803391179	"Note that \$p(x)\$ defines a covering \$p: \\mathbb{P}_x^1 \\rightarrow \\mathbb{P}^1_z\$ of de(...TRUNCATED)	https://epoch.ai/frontiermath/tiers-1-4/benchmark-problems	"ডিগ্ৰী 19-ৰ এনে এটা বহুপদ \$p(x) \\in \\mathbb{C}[x]\$ গঠ(...TRUNCATED)	"একটি 19 মাত্রার বহুপদী \$p(x) \\in \\mathbb{C}[x]\$ গঠন (...TRUNCATED)	"એક 19 ઘાત ધરાવતી બહુપદી \$p(x) \\in \\mathbb{C}[x]\$ ની ર(...TRUNCATED)	"घात 19 का एक ऐसा बहुपद \$p(x) \\in \\mathbb{C}[x]\$ निर्म(...TRUNCATED)	"\$X:=\\{p(x) = p(y)\\} \\subset \\mathbb{P}^1 \\times \\mathbb{P}^1\$ ಯು \$\\mathbb{C}\$ (...TRUNCATED)	"\$X:=\\{p(x) = p(y)\\} \\subset \\mathbb{P}^1 \\times \\mathbb{P}^1\$-ന് \$\\mathbb{C}\$-(...TRUNCATED)	"अशी एक कोटी 19 असणारी बहुपदी \\(p(x) \\in \\mathbb{C}[x]\(...TRUNCATED)	"\$\\mathbb{C}\$ मा \$X:=\\{p(x) = p(y)\\} \\subset \\mathbb{P}^1 \\times \\mathbb{P}^1\$ (...TRUNCATED)	"ଗୋଟିଏ degree 19 ବହୁପଦୀ \$p(x) \\in \\mathbb{C}[x]\$ ନିର୍ମାଣ (...TRUNCATED)	"ਇੱਕ ਡਿਗਰੀ 19 ਬਹੁਪਦ \$p(x) \\in \\mathbb{C}[x]\$ ਇਸ ਤਰ੍ਹਾ(...TRUNCATED)	"एकं 19 घातयुक्तं बहुपदं \$p(x) \\in \\mathbb{C}[x]\$ एव(...TRUNCATED)	"\$X:=\\{p(x) = p(y)\\} \\subset \\mathbb{P}^1 \\times \\mathbb{P}^1\$ ஆனது \\(\\mathbb{C}(...TRUNCATED)	"\$\\mathbb{C}[x]\$ లో \$p(x)\$ అనే డిగ్రీ 19 బహుపదిని న(...TRUNCATED)	"ایک درجہ 19 کا کثیر رقمی \$p(x) \\in \\mathbb{C}[x]\$ اس طرح تشکیل د(...TRUNCATED)
frontiermath-tier2-prime-field-continuous-extensions	2	Prime field continuous extensions	"Let \$a_n \$ for \$n \\in \\mathbb{Z} \$ be the sequence of integers satisfying the recurrence (...TRUNCATED)	9811	"The idea of the solution comes from the Skolem-Mahler-Lech theorem. Let \\(\\alpha_1,\\dots,\\alpha(...TRUNCATED)	https://epoch.ai/frontiermath/tiers-1-4/benchmark-problems	"ধৰা হওক \$n \\in \\mathbb{Z} \$-ৰ বাবে \$a_n \$ হৈছে পৌনঃ(...TRUNCATED)	"ধরি \$n \\in \\mathbb{Z} \$-এর জন্য \$a_n \$ হলো পূর্ণসং(...TRUNCATED)	"ધારો કે \$n \\in \\mathbb{Z} \$ માટે \$a_n \$ એ પુનરાવર્(...TRUNCATED)	"मान लीजिए कि \$n \\in \\mathbb{Z} \$ के लिए \$a_n \$ प्रा(...TRUNCATED)	"\$n \\in \\mathbb{Z} \$ ಗಾಗಿ \$a_n \$ ಎಂಬುದು $$0 \\leq i \\leq 3 $$ ಗೆ (...TRUNCATED)	"\$0 \\leq i \\leq 3 \$ എന്നതിനുള്ള പ്രാരംഭ നിബന്ധ(...TRUNCATED)	"समजा \$n \\in \\mathbb{Z} \$ साठी \$a_n \$ हा $$0 \\leq i \\leq 3 $$ सा(...TRUNCATED)	"मानौं \$n \\in \\mathbb{Z} \$ का लागि \$a_n \$ प्रारम्भ(...TRUNCATED)	"ମନେକରନ୍ତୁ \$n \\in \\mathbb{Z} \$ ପାଇଁ \$a_n \$ ହେଉଛି ନ(...TRUNCATED)	"ਮੰਨ ਲਓ \$n \\in \\mathbb{Z} \$ ਲਈ \$a_n \$ ਪੂਰਨ ਅੰਕਾਂ ਦਾ (...TRUNCATED)	"मन्यताम् \$n \\in \\mathbb{Z} \$ कृते \$a_n \$ इति पूर्(...TRUNCATED)	"\$n \\in \\mathbb{Z} \$-க்கு \$a_n \$ என்பது $$0 \\leq i \\leq 3 $$-க்(...TRUNCATED)	"\$n \\in \\mathbb{Z} \$ కొరకు \$a_n \$ అనేది\n\n\\[\na_n = 198130309625 a_{(...TRUNCATED)	"فرض کریں کہ \$n \\in \\mathbb{Z} \$ کے لیے \$a_n \$ صحیح اعداد کا ای(...TRUNCATED)
frontiermath-tier1-counting-nonzero-solutions-of-homogeneous-equations	1	Counting nonzero solutions of homogeneous equations	"How many nonzero points are there on\n[x^3y+y^3z+z^3x=0]\n\nover \$\\mathbb{F}_{5^{18}}\$ up to s(...TRUNCATED)	3814708984376	"Using the code presented below to count the solutions for \\(\\mathbb{F}_5,\\mathbb{F}_{25},\\mathb(...TRUNCATED)	https://epoch.ai/frontiermath/tiers-1-4/benchmark-problems	"স্কেলিং সাপেক্ষে \$\\mathbb{F}_{5^{18}}\$ -ৰ ওপৰত\n[x^3y+(...TRUNCATED)	"\$\\mathbb{F}_{5^{18}}\$ এর উপর স্কেলিং সাপেক্ষে\n[x^3y+y(...TRUNCATED)	"\$\\mathbb{F}_{5^{18}}\$ પર સ્કેલિંગ સુધી, \n[x^3y+y^3z+z^3x=0]\n\n(...TRUNCATED)	"\$\\mathbb{F}_{5^{18}}\$ पर स्केलिंग तक,\n[x^3y+y^3z+z^3x=0]\n\nपर (...TRUNCATED)	"\$\\mathbb{F}_{5^{18}}\$ ನ ಮೇಲೆ ಸ್ಕೇಲಿಂಗ್ ವರೆಗೆ\n[x^3y+y^3z(...TRUNCATED)	"\$\\mathbb{F}_{5^{18}}\$ -ന് മുകളിൽ സ്കെയിലിംഗ് വരെ ((...TRUNCATED)	"\$\\mathbb{F}_{5^{18}}\$ वर स्केलिंगपर्यंत (up to scaling)\n[x^3y(...TRUNCATED)	"\$\\mathbb{F}_{5^{18}}\$ माथि स्केलिङ सम्म\n[x^3y+y^3z+z^3x=0]\n\n(...TRUNCATED)	"\$\\mathbb{F}_{5^{18}}\$ ଉପରେ ସ୍କେଲିଂ ପର୍ଯ୍ୟନ୍ତ\n[x^3y+y^(...TRUNCATED)	"ਸਕੇਲਿੰਗ ਤੱਕ, \$\\mathbb{F}_{5^{18}}\$ ਉੱਤੇ\n[x^3y+y^3z+z^3x=0]\n\n'(...TRUNCATED)	"\$\\mathbb{F}_{5^{18}}\$ इत्यस्योपरि स्केलिंग-पर्य(...TRUNCATED)	"\$\\mathbb{F}_{5^{18}}\$ மீது அளவிடுதல் வரை (up to scaling),\n[x^(...TRUNCATED)	"\$\\mathbb{F}_{5^{18}}\$ పై స్కేలింగ్ వరకు\n[x^3y+y^3z+z^3x=0]\n\n(...TRUNCATED)	"\$\\mathbb{F}_{5^{18}}\$ پر اسکیلنگ کے لحاظ سے (up to scaling)\n[x^3y+y^3z+z^3x=(...TRUNCATED)

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indic-frontier-math

FrontierMath, from Epoch AI, is a benchmark of research-level mathematics problems in four tiers of difficulty, written to take expert mathematicians hours to weeks, each with a single automatically checkable answer. The benchmark itself is held privately; this dataset translates the 12 sample problems (tiers 1-4) that Epoch AI publishes on its website, which are too few to support a statistical claim.

indic-frontier-math is FrontierMath (Epoch AI's public sample problems) translated into 14 Indian languages by an LLM translation pipeline with automatic quality checks, with a portion of the translations reviewed and corrected by human annotators. It is part of the Indic-LLM-Eval-Suite.

Upstream source: https://epoch.ai/frontiermath/tiers-1-4/benchmark-problems

Languages

Assamese (as), Bengali (bn), Gujarati (gu), Hindi (hi), Kannada (kn), Malayalam (ml), Marathi (mr), Nepali (ne), Odia (or), Punjabi (pa), Sanskrit (sa), Tamil (ta), Telugu (te), Urdu (ur)

Layout

One row per original benchmark item, with every language on that same row. The original columns are carried through unchanged; each translated field gains one column per language:

problem -> problem_Hindi_translation, problem_Bengali_translation, ...


rows	12
languages	14
translated fields	`problem`
columns	21

More details

For how the translations were made and verified, and for model results on the suite, see the report and the blog:

Report: coming soon
Blog: Introducing Indic-LLM-Eval-Suite

Citation

If you use this dataset, please cite:

@misc{indicllmevalsuite2026blog,
  title        = {Introducing Indic-LLM-Eval-Suite: Better Indic Models Need Better Benchmarks},
  author       = {Kunchukuttan, Anoop and Kamath, Anusha and Khapra, Mitesh M. and
                  Khan, Mohammed Safi Ur Rahman and Agarwal, Niket and Thethy, Rajpreet and
                  Akkati, Ramani and Joshi, Raviraj and Murthy, Rudra and
                  Jayakumar, Thanmay and Vaidya, Utkarsh},
  year         = {2026},
  month        = oct,
  howpublished = {\url{https://ai4bharat.github.io/indic-llm-eval-suite-blog/}},
  note         = {AI4Bharat, IIT Madras and NVIDIA}
}

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Collection including ai4bharat/indic-frontier-math

Indic-LLM-Eval-Suite

Collection

An open, community-driven evolving collection of benchmarks for comprehensive and reliable evaluation of LLMs across Indian Languages • 23 items • Updated 3 days ago • 1

id large_stringlengths 48 71	tier int64 1 4	title large_stringlengths 29 52	problem large_stringlengths 100 1.68k	answer large_stringlengths 3 74	solution large_stringlengths 761 20k	source_url large_stringclasses 1 value	problem_Assamese_translation large_stringlengths 98 1.85k	problem_Bengali_translation large_stringlengths 99 1.85k	problem_Gujarati_translation large_stringlengths 90 1.96k	problem_Hindi_translation large_stringlengths 86 1.88k	problem_Kannada_translation large_stringlengths 102 1.87k	problem_Malayalam_translation large_stringlengths 133 2.01k	problem_Marathi_translation large_stringlengths 106 1.96k	problem_Nepali_translation large_stringlengths 95 1.84k	problem_Odia_translation large_stringlengths 99 1.78k	problem_Punjabi_translation large_stringlengths 92 1.86k	problem_Sanskrit_translation large_stringlengths 110 1.88k	problem_Tamil_translation large_stringlengths 120 1.87k	problem_Telugu_translation large_stringlengths 98 1.86k	problem_Urdu_translation large_stringlengths 107 1.85k
frontiermath-tier4-an-optimization-problem-in-bmo-space	4	An optimization problem in BMO space	Let \[ c = \sup_{f} \int_{0}^{1} (f(t)^{3}+\|f(t)\|)dt\] where supremum is taken over all integrable functions \(f : [0,1] \to \mathbb{R}\) such that \(\int_{0}^{1}f(t)dt=-10\), \(\int_{0}^{1} f^{2}(t)dt=100+\frac{1}{12}\), and \[ \sup_{J \subset [0,1]} \frac{1}{\|J\|}\int_{J} \left\|f(t) - \frac{1}{\|J\|}\int_{J}f(s)ds \...	\(\frac{\sqrt{3}}{36} + \frac{\sqrt{3}}{6} e^{-20\sqrt{3}-\frac{1}{6}}\)	Let us consider the following optimization problem \[\begin{align*} &B(x_{1}, x_{2}; \varepsilon) \stackrel{\mathrm{def}}{=}\\ &\sup_{f \in L^{1}([0,1])}\left\{\int_{0}^{1}F(f(t))dt\, : \, \int_{0}^{1} f(t)dt=x_{1}, \, \int_{0}^{1}f^{2}(t) dt=x_{2},\, \sup_{J \subset [0,1]} \frac{1}{\|J\|}\int_{J}\left\|f(t) - \frac{...	https://epoch.ai/frontiermath/tiers-1-4/benchmark-problems	ধৰা হওক \[ c = \sup_{f} \int_{0}^{1} (f(t)^{3}+\|f(t)\|)dt\] য’ত supremum লোৱা হৈছে এনে সকলো অনুকলনযোগ্য ফলন \(f : [0,1] \to \mathbb{R}\)-ৰ ওপৰত যাতে \(\int_{0}^{1}f(t)dt=-10\), \(\int_{0}^{1} f^{2}(t)dt=100+\frac{1}{12}\), আৰু \[ \sup_{J \subset [0,1]} \frac{1}{\|J\|}\int_{J} \left\|f(t) - \frac{1}{\|J\|}\int_{J}f(s)ds \...	ধরি, \[ c = \sup_{f} \int_{0}^{1} (f(t)^{3}+\|f(t)\|)dt\] যেখানে সুপ্রিমামটি এমন সকল সমাকলনযোগ্য ফাংশন \(f : [0,1] \to \mathbb{R}\)-এর উপর নেওয়া হয়েছে যেন \(\int_{0}^{1}f(t)dt=-10\), \(\int_{0}^{1} f^{2}(t)dt=100+\frac{1}{12}\), এবং \[ \sup_{J \subset [0,1]} \frac{1}{\|J\|}\int_{J} \left\|f(t) - \frac{1}{\|J\|}\int_{J}f...	ધારો કે \[ c = \sup_{f} \int_{0}^{1} (f(t)^{3}+\|f(t)\|)dt\] જ્યાં સુપ્રિમમ એવા તમામ સંકલનીય વિધેયો \(f : [0,1] \to \mathbb{R}\) પર લેવામાં આવે છે કે જેથી \(\int_{0}^{1}f(t)dt=-10\), \(\int_{0}^{1} f^{2}(t)dt=100+\frac{1}{12}\), અને \[ \sup_{J \subset [0,1]} \frac{1}{\|J\|}\int_{J} \left\|f(t) - \frac{1}{\|J\|}\int_{J}f(s...	मान लीजिए कि \[ c = \sup_{f} \int_{0}^{1} (f(t)^{3}+\|f(t)\|)dt\] जहाँ supremum ऐसे सभी समाकलनीय फलनों \(f : [0,1] \to \mathbb{R}\) पर लिया गया है जिनके लिए \(\int_{0}^{1}f(t)dt=-10\), \(\int_{0}^{1} f^{2}(t)dt=100+\frac{1}{12}\), और \[ \sup_{J \subset [0,1]} \frac{1}{\|J\|}\int_{J} \left\|f(t) - \frac{1}{\|J\|}\int_{J}f(...	\[ c = \sup_{f} \int_{0}^{1} (f(t)^{3}+\|f(t)\|)dt\] ಇರಲಿ, ಇಲ್ಲಿ ಸುಪ್ರಿಮಮ್ ಅನ್ನು \(\int_{0}^{1}f(t)dt=-10\), \(\int_{0}^{1} f^{2}(t)dt=100+\frac{1}{12}\), ಮತ್ತು \[ \sup_{J \subset [0,1]} \frac{1}{\|J\|}\int_{J} \left\|f(t) - \frac{1}{\|J\|}\int_{J}f(s)ds \right\|^{2}dt \leq \frac{1}{12},\] ಆಗಿರುವಂತಹ ಎಲ್ಲಾ ಸಮಾಕಲನೀಯ ಫಲನಗಳು \(...	\[ c = \sup_{f} \int_{0}^{1} (f(t)^{3}+\|f(t)\|)dt\] ആയിരിക്കട്ടെ, ഇവിടെ സുപ്രീമം എന്നത് \(\int_{0}^{1}f(t)dt=-10\), \(\int_{0}^{1} f^{2}(t)dt=100+\frac{1}{12}\), കൂടാതെ \[ \sup_{J \subset [0,1]} \frac{1}{\|J\|}\int_{J} \left\|f(t) - \frac{1}{\|J\|}\int_{J}f(s)ds \right\|^{2}dt \leq \frac{1}{12},\] എന്നിങ്ങനെയുള്ള എല്ലാ ഇന്...	समजा \[ c = \sup_{f} \int_{0}^{1} (f(t)^{3}+\|f(t)\|)dt\] येथे supremum अशा सर्व संकलनीय (integrable) फलनांवर \(f : [0,1] \to \mathbb{R}\) घेतला आहे ज्यांच्यासाठी \(\int_{0}^{1}f(t)dt=-10\), \(\int_{0}^{1} f^{2}(t)dt=100+\frac{1}{12}\), आणि \[ \sup_{J \subset [0,1]} \frac{1}{\|J\|}\int_{J} \left\|f(t) - \frac{1}{\|J\|}\in...	मानौँ \[ c = \sup_{f} \int_{0}^{1} (f(t)^{3}+\|f(t)\|)dt\] जहाँ supremum \(\int_{0}^{1}f(t)dt=-10\), \(\int_{0}^{1} f^{2}(t)dt=100+\frac{1}{12}\), र \[ \sup_{J \subset [0,1]} \frac{1}{\|J\|}\int_{J} \left\|f(t) - \frac{1}{\|J\|}\int_{J}f(s)ds \right\|^{2}dt \leq \frac{1}{12},\] हुने गरी सबै समाकलनीय (integrable) फलनहरू \(f...	ଧରାଯାଉ \[ c = \sup_{f} \int_{0}^{1} (f(t)^{3}+\|f(t)\|)dt\] ଯେଉଁଠାରେ ସୁପ୍ରିମମ୍ (supremum) ସେହି ସମସ୍ତ ସମାକଳନୀୟ ଫଳନ \(f : [0,1] \to \mathbb{R}\) ଗୁଡ଼ିକ ଉପରେ ନିଆଯାଇଛି ଯେପରିକି \(\int_{0}^{1}f(t)dt=-10\), \(\int_{0}^{1} f^{2}(t)dt=100+\frac{1}{12}\), ଏବଂ \[ \sup_{J \subset [0,1]} \frac{1}{\|J\|}\int_{J} \left\|f(t) - \frac{1...	ਮੰਨ ਲਓ \[ c = \sup_{f} \int_{0}^{1} (f(t)^{3}+\|f(t)\|)dt\] ਜਿੱਥੇ supremum ਉਹਨਾਂ ਸਾਰੇ integrable ਫੰਕਸ਼ਨਾਂ \(f : [0,1] \to \mathbb{R}\) ਉੱਤੇ ਲਿਆ ਗਿਆ ਹੈ ਤਾਂ ਜੋ \(\int_{0}^{1}f(t)dt=-10\), \(\int_{0}^{1} f^{2}(t)dt=100+\frac{1}{12}\), ਅਤੇ \[ \sup_{J \subset [0,1]} \frac{1}{\|J\|}\int_{J} \left\|f(t) - \frac{1}{\|J\|}\int_{J}...	अस्तु \[ c = \sup_{f} \int_{0}^{1} (f(t)^{3}+\|f(t)\|)dt\] यत्र उच्चिष्ठः (supremum) तादृशानां सर्वेषां समाकलनीय-फलनानां (integrable functions) \(f : [0,1] \to \mathbb{R}\) उपरि गृहीतः यैः \(\int_{0}^{1}f(t)dt=-10\), \(\int_{0}^{1} f^{2}(t)dt=100+\frac{1}{12}\), तथा \[ \sup_{J \subset [0,1]} \frac{1}{\|J\|}\int_{J} \le...	\[ c = \sup_{f} \int_{0}^{1} (f(t)^{3}+\|f(t)\|)dt\] என்க. இங்கு \(\int_{0}^{1}f(t)dt=-10\), \(\int_{0}^{1} f^{2}(t)dt=100+\frac{1}{12}\), மற்றும் \[ \sup_{J \subset [0,1]} \frac{1}{\|J\|}\int_{J} \left\|f(t) - \frac{1}{\|J\|}\int_{J}f(s)ds \right\|^{2}dt \leq \frac{1}{12},\] என்பவற்றை நிறைவு செய்யும் அனைத்து தொகையிடத்தக்க ...	\[ c = \sup_{f} \int_{0}^{1} (f(t)^{3}+\|f(t)\|)dt\] అనుకుందాం, ఇక్కడ సుప్రిమమ్ అనేది \(\int_{0}^{1}f(t)dt=-10\), \(\int_{0}^{1} f^{2}(t)dt=100+\frac{1}{12}\), మరియు \[ \sup_{J \subset [0,1]} \frac{1}{\|J\|}\int_{J} \left\|f(t) - \frac{1}{\|J\|}\int_{J}f(s)ds \right\|^{2}dt \leq \frac{1}{12},\] అయ్యేటట్లుగా ఉండే అన్ని సమాకల...	فرض کریں کہ \[ c = \sup_{f} \int_{0}^{1} (f(t)^{3}+\|f(t)\|)dt\] جہاں supremum ان تمام قابلِ تکامل تفاعلات \(f : [0,1] \to \mathbb{R}\) پر لیا گیا ہے جن کے لیے \(\int_{0}^{1}f(t)dt=-10\)، \(\int_{0}^{1} f^{2}(t)dt=100+\frac{1}{12}\)، اور \[ \sup_{J \subset [0,1]} \frac{1}{\|J\|}\int_{J} \left\|f(t) - \frac{1}{\|J\|}\int_{...
frontiermath-tier4-elliptic-curves-and-higher-galois-theory	4	Elliptic curves and higher Galois Theory	Define functions \(F(z)\) and \(G(z)\) on the upper-half of the complex plane by \(F(z):=1+\sum_{n=1}^{\infty} \frac{e^{2\pi i n^2z}}{\prod_{j=1}^n (1+e^{2 \pi i jz})^2} \ \ \ \textrm{and}\) \(G(z):=\prod_{n=1}^{\infty} \frac{(1-e^{2\pi i nz})(1-e^{2\pi i (2n-1)z})}{(1+e^{2\pi i n z})}.\) Let \(\ell_1\) be the smalles...	73948492097301691765464030714938921180352979774791349212169829996615700482	We first determine the pair \((\ell_1, \ell_2)\) by examining conditions (1)-(4). Conditions (1) and (2) are standard in algebraic number theory. Condition (1) demands (see Ch. 5.1 of [1]) that \(D_{\ell_1}:=-\ell_1\) is the discriminant of the ring of integers of the imaginary quadratic field \(\mathbb Q(\sqrt{-\e...	https://epoch.ai/frontiermath/tiers-1-4/benchmark-problems	জটিল সমতলৰ উচ্চ-অৰ্ধাংশত ফলন \(F(z)\) আৰু \(G(z)\) ক তলত দিয়া ধৰণে সংজ্ঞায়িত কৰক \(F(z):=1+\sum_{n=1}^{\infty} \frac{e^{2\pi i n^2z}}{\prod_{j=1}^n (1+e^{2 \pi i jz})^2} \ \ \ \textrm{আৰু}\) \(G(z):=\prod_{n=1}^{\infty} \frac{(1-e^{2\pi i nz})(1-e^{2\pi i (2n-1)z})}{(1+e^{2\pi i n z})}.\) ধৰক \(\ell_1\) হৈছে এনে আটাইত...	জটিল তলের ঊর্ধ্ব-অর্ধে (upper-half of the complex plane) \(F(z)\) এবং \(G(z)\) ফাংশন দুটি নিম্নোক্তভাবে সংজ্ঞায়িত করুন: \(F(z):=1+\sum_{n=1}^{\infty} \frac{e^{2\pi i n^2z}}{\prod_{j=1}^n (1+e^{2 \pi i jz})^2} \ \ \ \textrm{এবং}\) \(G(z):=\prod_{n=1}^{\infty} \frac{(1-e^{2\pi i nz})(1-e^{2\pi i (2n-1)z})}{(1+e^{2\pi i ...	સંકર સમતલના ઉપલા અર્ધભાગ (upper-half of the complex plane) પર વિધેયો \(F(z)\) અને \(G(z)\) ને નીચે પ્રમાણે વ્યાખ્યાયિત કરો: \(F(z):=1+\sum_{n=1}^{\infty} \frac{e^{2\pi i n^2z}}{\prod_{j=1}^n (1+e^{2 \pi i jz})^2} \ \ \ \textrm{અને}\) \(G(z):=\prod_{n=1}^{\infty} \frac{(1-e^{2\pi i nz})(1-e^{2\pi i (2n-1)z})}{(1+e^{2\pi...	सम्मिश्र तल के ऊपरी अर्ध-तल पर फलनों \(F(z)\) और \(G(z)\) को निम्न प्रकार परिभाषित कीजिए: \(F(z):=1+\sum_{n=1}^{\infty} \frac{e^{2\pi i n^2z}}{\prod_{j=1}^n (1+e^{2 \pi i jz})^2} \ \ \ \textrm{तथा}\) \(G(z):=\prod_{n=1}^{\infty} \frac{(1-e^{2\pi i nz})(1-e^{2\pi i (2n-1)z})}{(1+e^{2\pi i n z})}.\) मान लीजिए \(\ell_1\)...	ಸಂಕೀರ್ಣ ಸಮತಲದ ಮೇಲಿನ ಅರ್ಧಭಾಗದಲ್ಲಿ \(F(z)\) ಮತ್ತು \(G(z)\) ಉತ್ಪನ್ನಗಳನ್ನು ಈ ಕೆಳಗಿನಂತೆ ವ್ಯಾಖ್ಯಾನಿಸಿ: \(F(z):=1+\sum_{n=1}^{\infty} \frac{e^{2\pi i n^2z}}{\prod_{j=1}^n (1+e^{2 \pi i jz})^2} \ \ \ \textrm{and}\) \(G(z):=\prod_{n=1}^{\infty} \frac{(1-e^{2\pi i nz})(1-e^{2\pi i (2n-1)z})}{(1+e^{2\pi i n z})}.\) ಕೆಳಗಿನ ಎಲ್ಲ ಷ...	കോംപ്ലക്സ് പ്രതലത്തിന്റെ മുകൾ പകുതിയിൽ (upper-half of the complex plane) \(F(z)\), \(G(z)\) എന്നീ ഫംഗ്ഷനുകൾ താഴെ പറയുന്ന വിധം നിർവചിക്കുക: \(F(z):=1+\sum_{n=1}^{\infty} \frac{e^{2\pi i n^2z}}{\prod_{j=1}^n (1+e^{2 \pi i jz})^2} \ \ \ \textrm{and}\) \(G(z):=\prod_{n=1}^{\infty} \frac{(1-e^{2\pi i nz})(1-e^{2\pi i (2n-1)...	सम्मिश्र प्रतलाच्या वरील अर्धभागावर (upper-half of the complex plane) \(F(z)\) आणि \(G(z)\) ही फलने पुढीलप्रमाणे परिभाषित करा: \(F(z):=1+\sum_{n=1}^{\infty} \frac{e^{2\pi i n^2z}}{\prod_{j=1}^n (1+e^{2 \pi i jz})^2} \ \ \ \textrm{आणि}\) \(G(z):=\prod_{n=1}^{\infty} \frac{(1-e^{2\pi i nz})(1-e^{2\pi i (2n-1)z})}{(1+e^{2...	सम्मिश्र समतलको माथिल्लो आधा भागमा फलनहरू \(F(z)\) र \(G(z)\) लाई निम्नद्वारा परिभाषित गर्नुहोस्: \(F(z):=1+\sum_{n=1}^{\infty} \frac{e^{2\pi i n^2z}}{\prod_{j=1}^n (1+e^{2 \pi i jz})^2} \ \ \ \textrm{र}\) \(G(z):=\prod_{n=1}^{\infty} \frac{(1-e^{2\pi i nz})(1-e^{2\pi i (2n-1)z})}{(1+e^{2\pi i n z})}.\) मानौं \(\ell_1...	ଜଟିଳ ସମତଳର ଉପର-ଅର୍ଦ୍ଧାଂଶ (upper-half of the complex plane) ରେ ଫଳନ \(F(z)\) ଏବଂ \(G(z)\) କୁ ନିମ୍ନଲିଖିତ ଭାବରେ ସଂଜ୍ଞାୟିତ କରନ୍ତୁ: \(F(z):=1+\sum_{n=1}^{\infty} \frac{e^{2\pi i n^2z}}{\prod_{j=1}^n (1+e^{2 \pi i jz})^2} \ \ \ \textrm{ଏବଂ}\) \(G(z):=\prod_{n=1}^{\infty} \frac{(1-e^{2\pi i nz})(1-e^{2\pi i (2n-1)z})}{(1+e^{2\...	ਕੰਪਲੈਕਸ ਤਲ ਦੇ ਉੱਪਰਲੇ ਅੱਧ ਉੱਤੇ ਫ਼ੰਕਸ਼ਨਾਂ \(F(z)\) ਅਤੇ \(G(z)\) ਨੂੰ ਇਸ ਪ੍ਰਕਾਰ ਪਰਿਭਾਸ਼ਿਤ ਕਰੋ \(F(z):=1+\sum_{n=1}^{\infty} \frac{e^{2\pi i n^2z}}{\prod_{j=1}^n (1+e^{2 \pi i jz})^2} \ \ \ \textrm{ਅਤੇ}\) \(G(z):=\prod_{n=1}^{\infty} \frac{(1-e^{2\pi i nz})(1-e^{2\pi i (2n-1)z})}{(1+e^{2\pi i n z})}.\) ਮੰਨ ਲਓ \(\ell_1\) ਉਹ...	सम्मिश्रतलस्य उपरि-अर्धभागे फलने \(F(z)\) तथा \(G(z)\) अनयोः रूपेण परिभाषयत: \(F(z):=1+\sum_{n=1}^{\infty} \frac{e^{2\pi i n^2z}}{\prod_{j=1}^n (1+e^{2 \pi i jz})^2} \ \ \ \textrm{तथा}\) \(G(z):=\prod_{n=1}^{\infty} \frac{(1-e^{2\pi i nz})(1-e^{2\pi i (2n-1)z})}{(1+e^{2\pi i n z})}.\) मान्यताम् \(\ell_1\) सा लघुतमा अभ...	சிக்கலெண் தளத்தின் மேல் அரைத்தளத்தில் \(F(z)\) மற்றும் \(G(z)\) ஆகிய சார்புகளைப் பின்வருமாறு வரையறுக்கவும்: \(F(z):=1+\sum_{n=1}^{\infty} \frac{e^{2\pi i n^2z}}{\prod_{j=1}^n (1+e^{2 \pi i jz})^2} \ \ \ \textrm{மற்றும்}\) \(G(z):=\prod_{n=1}^{\infty} \frac{(1-e^{2\pi i nz})(1-e^{2\pi i (2n-1)z})}{(1+e^{2\pi i n z})}.\)...	సంకీర్ణ తలం యొక్క ఎగువ అర్ధభాగంపై ప్రమేయాలు \(F(z)\) మరియు \(G(z)\) లను ఈ క్రింది విధంగా నిర్వచించండి: \(F(z):=1+\sum_{n=1}^{\infty} \frac{e^{2\pi i n^2z}}{\prod_{j=1}^n (1+e^{2 \pi i jz})^2} \ \ \ \textrm{మరియు}\) \(G(z):=\prod_{n=1}^{\infty} \frac{(1-e^{2\pi i nz})(1-e^{2\pi i (2n-1)z})}{(1+e^{2\pi i n z})}.\) క్రిం...	کمپلیکس پلین کے بالائی نصف پر تفاعلات \(F(z)\) اور \(G(z)\) کی تعریف یوں کریں کہ \(F(z):=1+\sum_{n=1}^{\infty} \frac{e^{2\pi i n^2z}}{\prod_{j=1}^n (1+e^{2 \pi i jz})^2} \ \ \ \textrm{اور}\) \(G(z):=\prod_{n=1}^{\infty} \frac{(1-e^{2\pi i nz})(1-e^{2\pi i (2n-1)z})}{(1+e^{2\pi i n z})}.\) فرض کریں کہ \(\ell_1\) وہ سب ...
frontiermath-tier3-combinatorics-in-the-tsirelson-space	3	Combinatorics in the Tsirelson space	Background. The Tsirelson space is a Banach space introduced in 1974 as a counterexample to a longstanding conjecture in Banach space theory. It is a subspace of \(\ell^\infty\) described as follows. Let \(P_k\) denote the operator on \(\ell^\infty\) that modifies each \(\langle x_n:\, n \in \mathbb N \rangle \in \ell...	402653211	Let us begin by introducing some helpful shorthand. In the first half of this explanation, we will deal with quite a few sequences of the form \(\langle 0,0,\dots,0,0,b,b-\varepsilon,b-2\varepsilon,\dots,a+2\varepsilon,a+\varepsilon,a,0,0,0,\dots \rangle\) where \(0 < a < b < 1\). For convenience, let \(\langle a,b,(m,...	https://epoch.ai/frontiermath/tiers-1-4/benchmark-problems	পৃষ্ঠভূমি। Tsirelson space হৈছে এটা Banach space যাক Banach space তত্ত্বৰ এক দীৰ্ঘদিনীয়া অনুমানৰ এটা counterexample হিচাপে 1974 চনত প্ৰৱৰ্তন কৰা হৈছিল। ই হৈছে \(\ell^\infty\) ৰ এটা subspace যাক তলত দিয়া ধৰণে বৰ্ণনা কৰা হৈছে। ধৰা হওক \(P_k\) এ \(\ell^\infty\) ৰ ওপৰত এনে এক operator ক বুজায় যিয়ে প্ৰতিটো \(\langle x_...	প্রেক্ষাপট। Tsirelson স্পেস হলো একটি Banach স্পেস যা Banach স্পেস তত্ত্বের একটি দীর্ঘস্থায়ী অনুমানের পাল্টা উদাহরণ (counterexample) হিসেবে 1974 সালে প্রবর্তিত হয়েছিল। এটি \(\ell^\infty\)-এর একটি সাবস্পেস যা নিম্নরূপ বর্ণিত। ধরি \(P_k\) হলো \(\ell^\infty\)-এর ওপর এমন একটি অপারেটর যা প্রতিটি \(\langle x_n:\, n \in \ma...	પૃષ્ઠભૂમિ. સાયરેલસન અવકાશ (Tsirelson space) એ 1974 માં બનાખ અવકાશ સિદ્ધાંતમાં લાંબા સમયથી ચાલતી એક કલ્પનાના પ્રત્યુદાહરણ તરીકે રજૂ કરાયેલ એક બનાખ અવકાશ છે. તે \(\ell^\infty\) નો એક ઉપઅવકાશ છે જે નીચે મુજબ વર્ણવાયેલ છે. ધારો કે \(P_k\) એ \(\ell^\infty\) પરનો એવો પ્રક્રિયક દર્શાવે છે જે દરેક \(\langle x_n:\, n \in \math...	पृष्ठभूमि। त्सिरेलसन समष्टि (Tsirelson space) एक बानाख समष्टि (Banach space) है जिसे बानाख समष्टि सिद्धांत में एक दीर्घकालिक अनुमान (conjecture) के प्रत्युदाहरण (counterexample) के रूप में 1974 में प्रस्तुत किया गया था। यह \(\ell^\infty\) की एक उपसमष्टि है जिसे निम्नानुसार वर्णित किया गया है। मान लीजिए कि \(P_k\), \(\...	ಹಿನ್ನೆಲೆ. ಸಿರೆಲ್ಸನ್ ಸ್ಪೇಸ್ (Tsirelson space) ಎಂಬುದು ಬನಾಚ್ ಸ್ಪೇಸ್ ಸಿದ್ಧಾಂತದಲ್ಲಿನ ದೀರ್ಘಕಾಲದ ಊಹೆಗೆ (conjecture) ಪ್ರತಿದೃಷ್ಟಾಂತವಾಗಿ 1974 ರಲ್ಲಿ ಪರಿಚಯಿಸಲಾದ ಬನಾಚ್ ಸ್ಪೇಸ್ ಆಗಿದೆ. ಇದು ಕೆಳಗಿನಂತೆ ವಿವರಿಸಲಾದ \(\ell^\infty\) ನ ಒಂದು ಉಪಸ್ಪೇಸ್ ಆಗಿದೆ. \(P_k\) ಎಂಬುದು \(\ell^\infty\) ಮೇಲಿನ ಆಪರೇಟರ್ ಆಗಿರಲಿ, ಇದು ಪ್ರತಿಯೊಂದು \(\langle x_n:\, n ...	പശ്ചാത്തലം. ബനാക് സ്പേസ് തിയറിയിലെ (Banach space theory) ഒരു ചിരകാല അനുമാനത്തിന് ഒരു കൗണ്ടർ എക്സാമ്പിൾ (counterexample) ആയി 1974-ൽ അവതരിപ്പിക്കപ്പെട്ട ഒരു ബനാക് സ്പേസ് ആണ് സിറൽസൺ സ്പേസ് (Tsirelson space). ഇത് താഴെ വിവരിച്ചിരിക്കുന്ന പ്രകാരം \(\ell^\infty\)-ന്റെ ഒരു സബ്സ്പേസ് ആണ്. \(\ell^\infty\)-ലെ ഓരോ \(\langle x_n:\...	पार्श्वभूमी. त्सिरेलसन अवकाश (Tsirelson space) हे बनाख अवकाश सिद्धांतातील (Banach space theory) एका प्रदीर्घ काळापासून असलेल्या अनुमानाला प्रतिउदाहरण म्हणून 1974 मध्ये मांडलेले एक बनाख अवकाश (Banach space) आहे. हे खालीलप्रमाणे वर्णन केलेले \(\ell^\infty\) चे एक उपअवकाश (subspace) आहे. समजा \(P_k\) हे \(\ell^\infty\) व...	पृष्ठभूमि। Tsirelson space एक Banach space हो जुन सन् 1974 मा Banach space सिद्धान्तको एक लामो समयदेखिको अनुमानको प्रतिउदाहरण (counterexample) को रूपमा प्रस्तुत गरिएको थियो। यो \(\ell^\infty\) को एक subspace हो जसलाई निम्नानुसार वर्णन गरिएको छ। मानौँ \(P_k\) ले \(\ell^\infty\) मा रहेको यस्तो operator लाई जनाउँछ जसले प...	ପୃଷ୍ଠଭୂମି: Tsirelson space ହେଉଛି Banach space theory ର ଏକ ଦୀର୍ଘସ୍ଥାୟୀ ଅନୁମାନର ଏକ ପ୍ରତି-ଉଦାହରଣ (counterexample) ଭାବରେ 1974 ମସିହାରେ ପ୍ରବର୍ତ୍ତିତ ଏକ Banach space । ଏହା ନିମ୍ନରେ ବର୍ଣ୍ଣିତ \(\ell^\infty\) ର ଏକ ଉପସ୍ପେସ୍ (subspace) ଅଟେ । ମନେକର \(P_k\) ହେଉଛି \(\ell^\infty\) ଉପରେ ଏକ ଅପରେଟର୍ ଯାହା ପ୍ରଥମ \(k\) ଟି ସ୍ଥାନାଙ୍କକୁ \(0\) ସ...	ਪਿਛੋਕੜ। ਸਿਰੇਲਸਨ ਸਪੇਸ (Tsirelson space) ਇੱਕ ਬਾਨਾਕ ਸਪੇਸ (Banach space) ਹੈ ਜੋ 1974 ਵਿੱਚ ਬਾਨਾਕ ਸਪੇਸ ਸਿਧਾਂਤ ਵਿੱਚ ਇੱਕ ਲੰਬੇ ਸਮੇਂ ਤੋਂ ਚੱਲੇ ਆ ਰਹੇ ਅਨੁਮਾਨ ਦੇ ਖੰਡਨ-ਉਦਾਹਰਨ (counterexample) ਵਜੋਂ ਪੇਸ਼ ਕੀਤਾ ਗਿਆ ਸੀ। ਇਹ \(\ell^\infty\) ਦਾ ਇੱਕ ਸਬਸਪੇਸ ਹੈ ਜਿਸਦਾ ਵੇਰਵਾ ਹੇਠਾਂ ਦਿੱਤਾ ਗਿਆ ਹੈ। ਮੰਨ ਲਓ \(P_k\) \(\ell^\infty\) ਉੱਤੇ ਉਸ ਓਪਰੇਟਰ ਨੂੰ ਦਰ...	पृष्ठभूमिः। त्सिरेल्सन-समष्टिः (Tsirelson space) काचित् बानाख-समष्टिः अस्ति, या बानाख-समष्टि-सिद्धान्ते एकस्य चिरकालिकस्यानुमानस्य प्रत्युदाहरणरूपेण 1974 तमे वर्षे प्रवर्तिता। इयम् अधोलिखितरूपेण वर्णिता \(\ell^\infty\) इत्यस्य उपसमष्टिः अस्ति। मानयतु यत् \(P_k\) इति \(\ell^\infty\) इत्यस्मिन् सः संकारकः अस्ति, यः प्रत...	பின்னணி. Tsirelson வெளி என்பது Banach வெளி கோட்பாட்டில் நீண்டகாலமாக இருந்த ஒரு கணித ஊகத்திற்கு எதிர்ப்புதாரணமாக 1974-இல் அறிமுகப்படுத்தப்பட்ட ஒரு Banach வெளி ஆகும். இது பின்வருமாறு விவரிக்கப்படும் \(\ell^\infty\)-ன் ஓர் உள்வெளி ஆகும். \(P_k\) என்பது \(\ell^\infty\)-இல் உள்ள ஒவ்வோர் \(\langle x_n:\, n \in \mathbb N \ra...	నేపథ్యం. సిరెల్సన్ స్పేస్ (Tsirelson space) అనేది బనాచ్ స్పేస్ సిద్ధాంతంలో చాలా కాలంగా ఉన్న ఒక ఊహనానికి (conjecture) ప్రతికూల ఉదాహరణగా 1974లో ప్రవేశపెట్టబడిన ఒక బనాచ్ స్పేస్. ఇది క్రింది విధంగా వివరించబడిన \(\ell^\infty\) యొక్క ఉపస్పేస్. \(\ell^\infty\) పై \(P_k\) అనేది ప్రతి \(\langle x_n:\, n \in \mathbb N \rangle \...	پس منظر۔ سیریلسن اسپیس (Tsirelson space) ایک باناخ اسپیس (Banach space) ہے جو 1974 میں باناخ اسپیس تھیوری کے ایک دیرینہ مفروضے (conjecture) کی جوابی مثال (counterexample) کے طور پر متعارف کرائی گئی تھی۔ یہ \(\ell^\infty\) کی ایک ذیلی فضا (subspace) ہے جسے درج ذیل کے طور پر بیان کیا جاتا ہے۔ فرض کریں کہ \(P_k\)، \(\ell...
frontiermath-tier3-enumerative-geometry-over-finite-fields	3	Enumerative geometry over finite fields	Let \(U\subset\mathbb{P}H^0(\mathbb{P}^2_{\mathbb{Z}}, \mathcal{O}(2))\) be the space of smooth conics in \(\mathbb{P}^2_{\mathbb{Z}}\), and let \(Z\subset U^6\) be the closed subscheme parametrizing \(6\)-tuples \((C_1, \cdots, C_6)\) with \(C_1\) tangent to \(C_2, \cdots, C_6\). Let \(\pi: Z\to U^5\) be the map induc...	866	The limit \(L\) is \(\sum_{m=1}^{3264}\frac{1}{m} \approx 8.668,\) giving an answer of 866. To see this, we use Harris’s result (see “Galois groups of enumerative problems,” from Duke Math J. 1979) that the map \(\pi: Z\to V\) is degree \(3264\) (known classically) with Galois group the full symmetric group \(S_{3264}\...	https://epoch.ai/frontiermath/tiers-1-4/benchmark-problems	ধৰা হওক \(U\subset\mathbb{P}H^0(\mathbb{P}^2_{\mathbb{Z}}, \mathcal{O}(2))\) হ'ল \(\mathbb{P}^2_{\mathbb{Z}}\) -ত মসৃণ শংকুছেদসমূহৰ স্থান, আৰু ধৰা হওক \(Z\subset U^6\) হ'ল এনে \(6\)-টাপল \((C_1, \cdots, C_6)\) পেৰামিটাৰাইজ কৰা বন্ধ উপ-স্কিম (closed subscheme) য'ত \(C_1\) হ'ল \(C_2, \cdots, C_6\)-ৰ স্পৰ্শক। ধৰা হওক \(\p...	ধরি \(U\subset\mathbb{P}H^0(\mathbb{P}^2_{\mathbb{Z}}, \mathcal{O}(2))\) হলো \(\mathbb{P}^2_{\mathbb{Z}}\)-এ মসৃণ কনিকের স্পেস, এবং ধরি \(Z\subset U^6\) হলো এমন একটি বদ্ধ সাবস্কিম যা \(C_1\) কনিকটি \(C_2, \cdots, C_6\)-এর সাথে স্পর্শক বিশিষ্ট \(6\)-টিউপল \((C_1, \cdots, C_6)\)-সমূহকে প্যারামিটারাইজ করে। ধরি \(\pi: Z\t...	ધારો કે \(U\subset\mathbb{P}H^0(\mathbb{P}^2_{\mathbb{Z}}, \mathcal{O}(2))\) એ \(\mathbb{P}^2_{\mathbb{Z}}\) માં સ્મૂધ કોનિક્સ (smooth conics) ની સ્પેસ છે, અને ધારો કે \(Z\subset U^6\) એ \(C_1\) એ \(C_2, \cdots, C_6\) ને સ્પર્શક હોય તેવી \(6\)-ટપલ્સ \((C_1, \cdots, C_6)\) ને પેરામીટરાઇઝ કરતું ક્લોઝ્ડ સબસ્કીમ (closed su...	मान लीजिए कि \(U\subset\mathbb{P}H^0(\mathbb{P}^2_{\mathbb{Z}}, \mathcal{O}(2))\), \(\mathbb{P}^2_{\mathbb{Z}}\) में स्मूथ शांकवों (smooth conics) की समष्टि है, और मान लीजिए कि \(Z\subset U^6\) ऐसे \(6\)-टुपलों \((C_1, \cdots, C_6)\) को प्राचलीकृत करने वाली संवृत उप-स्कीम (closed subscheme) है जिनमें \(C_1\), \(C_2, \c...	\(U\subset\mathbb{P}H^0(\mathbb{P}^2_{\mathbb{Z}}, \mathcal{O}(2))\) ಎಂಬುದು \(\mathbb{P}^2_{\mathbb{Z}}\) ನಲ್ಲಿನ ನುಣುಪಾದ ಶಂಕುಜಗಳ (smooth conics) ಆಕಾಶವಾಗಿರಲಿ, ಮತ್ತು \(Z\subset U^6\) ಎಂಬುದು \(C_2, \cdots, C_6\) ಗಳಿಗೆ \(C_1\) ಸ್ಪರ್ಶಕವಾಗಿರುವ \(6\)-ಟಪಲ್‌ಗಳನ್ನು \((C_1, \cdots, C_6)\) ಪ್ಯಾರಾಮೀಟರೀಕರಿಸುವ ಮುಚ್ಚಿದ ಉಪಯೋಜನೆಯಾಗಿರಲಿ ...	\(\mathbb{P}^2_{\mathbb{Z}}\) ലെ smooth conics-ന്റെ space \(U\subset\mathbb{P}H^0(\mathbb{P}^2_{\mathbb{Z}}, \mathcal{O}(2))\) ആയിരിക്കട്ടെ, കൂടാതെ \(C_1\) എന്നത് \(C_2, \cdots, C_6\) എന്നിവയിലേക്ക് tangent ആയിട്ടുള്ള \(6\)-tuples \((C_1, \cdots, C_6)\)-നെ parametrise ചെയ്യുന്ന closed subscheme \(Z\subset U^6\) ആയിരിക്...	समजा \(U\subset\mathbb{P}H^0(\mathbb{P}^2_{\mathbb{Z}}, \mathcal{O}(2))\) ही \(\mathbb{P}^2_{\mathbb{Z}}\) मधील smooth conics ची स्पेस आहे, आणि समजा \(Z\subset U^6\) ही \(C_1\) हे \(C_2, \cdots, C_6\) ला tangent असणाऱ्या \(6\)-tuples \((C_1, \cdots, C_6)\) चे प्राचलनीकरण करणारी closed subscheme आहे. समजा \(\pi: Z\to U^...	मानौं \(U\subset\mathbb{P}H^0(\mathbb{P}^2_{\mathbb{Z}}, \mathcal{O}(2))\), \(\mathbb{P}^2_{\mathbb{Z}}\) मा स्मूथ शंकुहरूको स्पेस हो, र मानौं \(Z\subset U^6\) भन्नाले \(C_1\) लाई \(C_2, \cdots, C_6\) सँग स्पर्शरेखीय हुने गरी \(6\)-tuples \((C_1, \cdots, C_6)\) लाई परिमितीकरण गर्ने खुला उपयोजना हो। मानौं \(\pi: Z\to U^...	ମନେକର \(U\subset\mathbb{P}H^0(\mathbb{P}^2_{\mathbb{Z}}, \mathcal{O}(2))\) ହେଉଛି \(\mathbb{P}^2_{\mathbb{Z}}\) ରେ smooth conics ର space, ଏବଂ ମନେକର \(Z\subset U^6\) ହେଉଛି \(C_1\) ଟି \(C_2, \cdots, C_6\) ସହିତ tangent ଥିବା \(6\)-tuples \((C_1, \cdots, C_6)\) କୁ parametrize କରୁଥିବା closed subscheme। ମନେକର \(\pi: Z\to U^5\)...	ਮੰਨ ਲਓ ਕਿ \(U\subset\mathbb{P}H^0(\mathbb{P}^2_{\mathbb{Z}}, \mathcal{O}(2))\), \(\mathbb{P}^2_{\mathbb{Z}}\) ਵਿੱਚ ਸਮੂਥ ਕੌਨਿਕਾਂ (smooth conics) ਦੀ ਸਪੇਸ ਹੈ, ਅਤੇ ਮੰਨ ਲਓ ਕਿ \(Z\subset U^6\) ਅਜਿਹੇ \(6\)-ਟਪਲਜ਼ \((C_1, \cdots, C_6)\) ਨੂੰ ਪੈਰਾਮੀਟਰਾਈਜ਼ ਕਰਨ ਵਾਲੀ ਕਲੋਜ਼ਡ ਸਬ-ਸਕੀਮ (closed subscheme) ਹੈ ਜਿਨ੍ਹਾਂ ਵਿੱਚ \(C_1\), \(C_2, ...	मन्यताम् \(U\subset\mathbb{P}H^0(\mathbb{P}^2_{\mathbb{Z}}, \mathcal{O}(2))\) \(\mathbb{P}^2_{\mathbb{Z}}\) इत्यस्मिन् मसृणशाङ्कवानाम् अवकाशः अस्ति, तथा च \(Z\subset U^6\) सा संवृतोपयोजना (closed subscheme) अस्ति या तादृशान् \(6\)-गणान् \((C_1, \cdots, C_6)\) प्राचलीकरोति यत्र \(C_1\) इत्ययं \(C_2, \cdots, C_6\) इत्येत...	\(U\subset\mathbb{P}H^0(\mathbb{P}^2_{\mathbb{Z}}, \mathcal{O}(2))\) என்பது \(\mathbb{P}^2_{\mathbb{Z}}\) இல் உள்ள மென்மையான கூம்புவெட்டுகளின் (smooth conics) வெளி என்க, மற்றும் \(Z\subset U^6\) என்பது \(C_2, \cdots, C_6\) உடன் \(C_1\) தொடுநிலையில் உள்ளவாறான \(6\)-வரிசைத் தொகுப்புகளை (\(6\)-tuples) \((C_1, \cdots, C_6)...	\(U\subset\mathbb{P}H^0(\mathbb{P}^2_{\mathbb{Z}}, \mathcal{O}(2))\) అనేది \(\mathbb{P}^2_{\mathbb{Z}}\) లోని smooth conics యొక్క ప్రదేశం (space) అనుకుందాం, మరియు \(Z\subset U^6\) అనేది \(C_2, \cdots, C_6\) లకు \(C_1\) tangent గా ఉండే \(6\)-tuples \((C_1, \cdots, C_6)\) లను పారామితికరించే closed subscheme అనుకుందాం. \(...	فرض کریں کہ \(U\subset\mathbb{P}H^0(\mathbb{P}^2_{\mathbb{Z}}, \mathcal{O}(2))\)، \(\mathbb{P}^2_{\mathbb{Z}}\) میں ہموار مخروطات (smooth conics) کی فضا ہے، اور فرض کریں کہ \(Z\subset U^6\) ایک بند ذیلی اسکیم (closed subscheme) ہے جو ایسے \(6\)-tuples \((C_1, \cdots, C_6)\) کو پیرامیٹرائز کرتی ہے جن میں \(C_1\)، مخروطا...
frontiermath-tier3-testing-artin-s-primitive-root-conjecture	3	Testing Artin’s primitive root conjecture	"For a positive integer \\(n\\), let \\(v_p(n)\\) denote the largest integer \\(v\\) such that \\(p^(...TRUNCATED)	367707	"We will break this into multiple sections.\n\nBackground. First of all, the presence of the \\(x\\)(...TRUNCATED)	https://epoch.ai/frontiermath/tiers-1-4/benchmark-problems	"এটা ধনাত্মক অখণ্ড সংখ্যা \\(n\\)-ৰ বাবে, ধৰ(...TRUNCATED)	"একটি ধনাত্মক পূর্ণসংখ্যা \\(n\\)-এর জন্য, (...TRUNCATED)	"ધન પૂર્ણાંક \\(n\\) માટે, ધારો કે \\(v_p(n)\\) એ એવ(...TRUNCATED)	"एक धनात्मक पूर्णांक \\(n\\) के लिए, माना \\(v_p(...TRUNCATED)	"ಧನಾತ್ಮಕ ಪೂರ್ಣಾಂಕ \\(n\\) ಗಾಗಿ, \\(p^v \\mid n\\) ಆಗಿರ(...TRUNCATED)	"ഒരു പോസിറ്റീവ് പൂർണ്ണസംഖ്യ \\(n\\)-ന്, \\(p^v \(...TRUNCATED)	"एका धन पूर्णांक \\(n\\) साठी, \\(v_p(n)\\) हा असा स(...TRUNCATED)	"एक धनात्मक पूर्णाङ्क \\(n\\) का लागि, मानौ(...TRUNCATED)	"ଏକ ଧନାତ୍ମକ ପୂର୍ଣ୍ଣସଂଖ୍ୟା \\(n\\) ପାଇଁ, ମନ(...TRUNCATED)	"ਇੱਕ ਧਨਾਤਮਕ ਪੂਰਨ ਅੰਕ \\(n\\) ਲਈ, ਮੰਨ ਲਓ \\(v_p(n)\\) (...TRUNCATED)	"कस्मैचित् धनात्मकपूर्णाङ्काय \\(n\\), \\(v_p((...TRUNCATED)	"ஒரு மிகை முழு எண் \\(n\\)-க்கு, \\(p^v \\mid n\\) என்ற(...TRUNCATED)	"ఒక ధన పూర్ణాంకం \\(n\\) కొరకు, \\(p^v \\mid n\\) అయ్య(...TRUNCATED)	"ایک مثبت صحیح عدد \\(n\\) کے لیے، فرض کریں کہ \\(v_p(n)\\) اس سب (...TRUNCATED)
frontiermath-tier2-a-recursive-construction-on-large-permutations	2	A recursive construction on large permutations	"Let \\(W\\) be the set of finite words with all distinct letters over the alphabet of positive inte(...TRUNCATED)	499999999972	"For \\(w\\in W\\), write \\(x\\prec_w y\\) to mean \\(x\\) appears before \\(y\\) in \\(w\\).\nSupp(...TRUNCATED)	https://epoch.ai/frontiermath/tiers-1-4/benchmark-problems	"ধৰা হওক \\(W\\) হৈছে ধনাত্মক অখণ্ড সংখ্যা(...TRUNCATED)	"ধরি \\(W\\) হলো ধনাত্মক পূর্ণসংখ্যার বর্ণ(...TRUNCATED)	"ધારો કે \\(W\\) એ ધન પૂર્ણાંકોના મૂળાક્ષર(...TRUNCATED)	"मान लीजिए कि \\(W\\) धनात्मक पूर्णांकों क(...TRUNCATED)	"\\(W\\) ಎಂಬುದು ಧನಾತ್ಮಕ ಪೂರ್ಣಾಂಕಗಳ ವರ್ಣಮಾ(...TRUNCATED)	"പോസിറ്റീവ് പൂർണ്ണസംഖ്യകളുടെ ആൽഫബെറ(...TRUNCATED)	"समजा \\(W\\) हा धन पूर्णांकांच्या वर्णमाल(...TRUNCATED)	"मानौं \\(W\\) धनात्मक पूर्णाङ्कहरूको वर्(...TRUNCATED)	"ମନେକରନ୍ତୁ \\(W\\) ହେଉଛି ଧନାତ୍ମକ ପୂର୍ଣ୍ଣସ(...TRUNCATED)	"ਮੰਨ ਲਓ \\(W\\) ਧਨਾਤਮਕ ਪੂਰਨ ਅੰਕਾਂ ਦੀ ਵਰਣਮਾਲ(...TRUNCATED)	"धनात्मकपूर्णाङ्कानां वर्णमालायां स(...TRUNCATED)	"நேர்ம முழு எண்களின் அகரவரிசையின் மீ(...TRUNCATED)	"ధన పూర్ణాంకాల వర్ణమాల (alphabet) పై అన్ని వ(...TRUNCATED)	"فرض کریں کہ \\(W\\) مثبت صحیح اعداد کے حروف تہجی پر تمام مم(...TRUNCATED)
frontiermath-tier2-asymptotics-of-solutions-of-a-5-variable-equation	2	Asymptotics of solutions of a 5-variable equation	"There are unique constants \\(C, \\alpha, \\beta\\) such that the number of solutions to the equati(...TRUNCATED)	214	"This count is equal to\n\\( \\sum_{n\\le x}d_{2}(n)d_{3}(n+1)\\)\nwhere \\(d_{k}(n) =\\displaystyle(...TRUNCATED)	https://epoch.ai/frontiermath/tiers-1-4/benchmark-problems	"এনে অনন্য ধ্ৰুৱক \\(C, \\alpha, \\beta\\) আছে যাতে \\(a, (...TRUNCATED)	"এমন অনন্য ধ্রুবক \\(C, \\alpha, \\beta\\) রয়েছে যেন \\((...TRUNCATED)	"એવા અનન્ય અચળાંકો \\(C, \\alpha, \\beta\\) છે કે જેથી(...TRUNCATED)	"ऐसे अद्वितीय अचर \\(C, \\alpha, \\beta\\) विद्यमान ह(...TRUNCATED)	"\\(a, b, c, d, e \\in \\mathbb{N}\\) ಮತ್ತು \\(ab \\le x\\) ಆಗಿರುವಾಗ\n\n\\(...TRUNCATED)	"\\(ab \\le x\\) ആയ \\(a, b, c, d, e \\in \\mathbb{N}\\) സഹിതമുള്ള\n\n\\(ab+1=(...TRUNCATED)	"\\(a, b, c, d, e \\in \\mathbb{N}\\) आणि \\(ab \\le x\\) सह\n\n\\(ab+1=cde\\)\n\nया (...TRUNCATED)	"त्यस्ता अद्वितीय अचरहरू \\(C, \\alpha, \\beta\\) छन(...TRUNCATED)	"ଏପରି ଅନନ୍ୟ ଧ୍ରୁବକ \\(C, \\alpha, \\beta\\) ରହିଛନ୍ତି (...TRUNCATED)	"ਅਜਿਹੇ ਵਿਲੱਖਣ ਅਚੱਲ \\(C, \\alpha, \\beta\\) ਮੌਜੂਦ ਹਨ ਕ(...TRUNCATED)	"एतादृशाः अनन्याः स्थिराङ्काः \\(C, \\alpha, \\b(...TRUNCATED)	"\\(x\\rightarrow\\infty\\) ஆகும்போது, \\(ab \\le x\\) என்றவாறுள(...TRUNCATED)	"\\(a, b, c, d, e \\in \\mathbb{N}\\) మరియు \\(ab \\le x\\) తో\n\n\\(ab+1=cde\\)\n\n(...TRUNCATED)	"ایسے منفرد مستقلات \\(C, \\alpha, \\beta\\) موجود ہیں کہ مساوات\n\n(...TRUNCATED)
frontiermath-tier2-find-the-degree-19-polynomial	2	Find the degree 19 polynomial	"Construct a degree 19 polynomial \\(p(x) \\in \\mathbb{C}[x]\\) such that \\(X:=\\{p(x) = p(y)\\} \(...TRUNCATED)	1876572071974094803391179	"Note that \\(p(x)\\) defines a covering \\(p: \\mathbb{P}_x^1 \\rightarrow \\mathbb{P}^1_z\\) of de(...TRUNCATED)	https://epoch.ai/frontiermath/tiers-1-4/benchmark-problems	"ডিগ্ৰী 19-ৰ এনে এটা বহুপদ \\(p(x) \\in \\mathbb{C}[x]\\) গঠ(...TRUNCATED)	"একটি 19 মাত্রার বহুপদী \\(p(x) \\in \\mathbb{C}[x]\\) গঠন (...TRUNCATED)	"એક 19 ઘાત ધરાવતી બહુપદી \\(p(x) \\in \\mathbb{C}[x]\\) ની ર(...TRUNCATED)	"घात 19 का एक ऐसा बहुपद \\(p(x) \\in \\mathbb{C}[x]\\) निर्म(...TRUNCATED)	"\\(X:=\\{p(x) = p(y)\\} \\subset \\mathbb{P}^1 \\times \\mathbb{P}^1\\) ಯು \\(\\mathbb{C}\\) (...TRUNCATED)	"\\(X:=\\{p(x) = p(y)\\} \\subset \\mathbb{P}^1 \\times \\mathbb{P}^1\\)-ന് \\(\\mathbb{C}\\)-(...TRUNCATED)	"अशी एक कोटी 19 असणारी बहुपदी \\(p(x) \\in \\mathbb{C}[x]\(...TRUNCATED)	"\\(\\mathbb{C}\\) मा \\(X:=\\{p(x) = p(y)\\} \\subset \\mathbb{P}^1 \\times \\mathbb{P}^1\\) (...TRUNCATED)	"ଗୋଟିଏ degree 19 ବହୁପଦୀ \\(p(x) \\in \\mathbb{C}[x]\\) ନିର୍ମାଣ (...TRUNCATED)	"ਇੱਕ ਡਿਗਰੀ 19 ਬਹੁਪਦ \\(p(x) \\in \\mathbb{C}[x]\\) ਇਸ ਤਰ੍ਹਾ(...TRUNCATED)	"एकं 19 घातयुक्तं बहुपदं \\(p(x) \\in \\mathbb{C}[x]\\) एव(...TRUNCATED)	"\\(X:=\\{p(x) = p(y)\\} \\subset \\mathbb{P}^1 \\times \\mathbb{P}^1\\) ஆனது \\(\\mathbb{C}(...TRUNCATED)	"\\(\\mathbb{C}[x]\\) లో \\(p(x)\\) అనే డిగ్రీ 19 బహుపదిని న(...TRUNCATED)	"ایک درجہ 19 کا کثیر رقمی \\(p(x) \\in \\mathbb{C}[x]\\) اس طرح تشکیل د(...TRUNCATED)
frontiermath-tier2-prime-field-continuous-extensions	2	Prime field continuous extensions	"Let \\(a_n \\) for \\(n \\in \\mathbb{Z} \\) be the sequence of integers satisfying the recurrence (...TRUNCATED)	9811	"The idea of the solution comes from the Skolem-Mahler-Lech theorem. Let \\(\\alpha_1,\\dots,\\alpha(...TRUNCATED)	https://epoch.ai/frontiermath/tiers-1-4/benchmark-problems	"ধৰা হওক \\(n \\in \\mathbb{Z} \\)-ৰ বাবে \\(a_n \\) হৈছে পৌনঃ(...TRUNCATED)	"ধরি \\(n \\in \\mathbb{Z} \\)-এর জন্য \\(a_n \\) হলো পূর্ণসং(...TRUNCATED)	"ધારો કે \\(n \\in \\mathbb{Z} \\) માટે \\(a_n \\) એ પુનરાવર્(...TRUNCATED)	"मान लीजिए कि \\(n \\in \\mathbb{Z} \\) के लिए \\(a_n \\) प्रा(...TRUNCATED)	"\\(n \\in \\mathbb{Z} \\) ಗಾಗಿ \\(a_n \\) ಎಂಬುದು $$0 \\leq i \\leq 3 $$ ಗೆ (...TRUNCATED)	"\\(0 \\leq i \\leq 3 \\) എന്നതിനുള്ള പ്രാരംഭ നിബന്ധ(...TRUNCATED)	"समजा \\(n \\in \\mathbb{Z} \\) साठी \\(a_n \\) हा $$0 \\leq i \\leq 3 $$ सा(...TRUNCATED)	"मानौं \\(n \\in \\mathbb{Z} \\) का लागि \\(a_n \\) प्रारम्भ(...TRUNCATED)	"ମନେକରନ୍ତୁ \\(n \\in \\mathbb{Z} \\) ପାଇଁ \\(a_n \\) ହେଉଛି ନ(...TRUNCATED)	"ਮੰਨ ਲਓ \\(n \\in \\mathbb{Z} \\) ਲਈ \\(a_n \\) ਪੂਰਨ ਅੰਕਾਂ ਦਾ (...TRUNCATED)	"मन्यताम् \\(n \\in \\mathbb{Z} \\) कृते \\(a_n \\) इति पूर्(...TRUNCATED)	"\\(n \\in \\mathbb{Z} \\)-க்கு \\(a_n \\) என்பது $$0 \\leq i \\leq 3 $$-க்(...TRUNCATED)	"\\(n \\in \\mathbb{Z} \\) కొరకు \\(a_n \\) అనేది\n\n\\[\na_n = 198130309625 a_{(...TRUNCATED)	"فرض کریں کہ \\(n \\in \\mathbb{Z} \\) کے لیے \\(a_n \\) صحیح اعداد کا ای(...TRUNCATED)
frontiermath-tier1-counting-nonzero-solutions-of-homogeneous-equations	1	Counting nonzero solutions of homogeneous equations	"How many nonzero points are there on\n[x^3y+y^3z+z^3x=0]\n\nover \\(\\mathbb{F}_{5^{18}}\\) up to s(...TRUNCATED)	3814708984376	"Using the code presented below to count the solutions for \\(\\mathbb{F}_5,\\mathbb{F}_{25},\\mathb(...TRUNCATED)	https://epoch.ai/frontiermath/tiers-1-4/benchmark-problems	"স্কেলিং সাপেক্ষে \\(\\mathbb{F}_{5^{18}}\\) -ৰ ওপৰত\n[x^3y+(...TRUNCATED)	"\\(\\mathbb{F}_{5^{18}}\\) এর উপর স্কেলিং সাপেক্ষে\n[x^3y+y(...TRUNCATED)	"\\(\\mathbb{F}_{5^{18}}\\) પર સ્કેલિંગ સુધી, \n[x^3y+y^3z+z^3x=0]\n\n(...TRUNCATED)	"\\(\\mathbb{F}_{5^{18}}\\) पर स्केलिंग तक,\n[x^3y+y^3z+z^3x=0]\n\nपर (...TRUNCATED)	"\\(\\mathbb{F}_{5^{18}}\\) ನ ಮೇಲೆ ಸ್ಕೇಲಿಂಗ್ ವರೆಗೆ\n[x^3y+y^3z(...TRUNCATED)	"\\(\\mathbb{F}_{5^{18}}\\) -ന് മുകളിൽ സ്കെയിലിംഗ് വരെ ((...TRUNCATED)	"\\(\\mathbb{F}_{5^{18}}\\) वर स्केलिंगपर्यंत (up to scaling)\n[x^3y(...TRUNCATED)	"\\(\\mathbb{F}_{5^{18}}\\) माथि स्केलिङ सम्म\n[x^3y+y^3z+z^3x=0]\n\n(...TRUNCATED)	"\\(\\mathbb{F}_{5^{18}}\\) ଉପରେ ସ୍କେଲିଂ ପର୍ଯ୍ୟନ୍ତ\n[x^3y+y^(...TRUNCATED)	"ਸਕੇਲਿੰਗ ਤੱਕ, \\(\\mathbb{F}_{5^{18}}\\) ਉੱਤੇ\n[x^3y+y^3z+z^3x=0]\n\n'(...TRUNCATED)	"\\(\\mathbb{F}_{5^{18}}\\) इत्यस्योपरि स्केलिंग-पर्य(...TRUNCATED)	"\\(\\mathbb{F}_{5^{18}}\\) மீது அளவிடுதல் வரை (up to scaling),\n[x^(...TRUNCATED)	"\\(\\mathbb{F}_{5^{18}}\\) పై స్కేలింగ్ వరకు\n[x^3y+y^3z+z^3x=0]\n\n(...TRUNCATED)	"\\(\\mathbb{F}_{5^{18}}\\) پر اسکیلنگ کے لحاظ سے (up to scaling)\n[x^3y+y^3z+z^3x=(...TRUNCATED)