id int64 41 3.22M | problem stringlengths 38 4.44k | solution stringlengths 8 38.6k | problem_vector listlengths 1.02k 1.02k | solution_vector listlengths 1.02k 1.02k | last_modified stringdate 2025-08-11 00:00:00 2025-08-11 00:00:00 |
|---|---|---|---|---|---|
599,349 | Let $P$ be a point in the interior of an acute triangle $ABC$, and let $Q$ be its isogonal conjugate. Denote by $\omega_P$ and $\omega_Q$ the circumcircles of triangles $BPC$ and $BQC$, respectively. Suppose the circle with diameter $\overline{AP}$ intersects $\omega_P$ again at $M$, and line $AM$ intersects $\omega_... | [b][color=red]Claim:[/color][/b] $M$ and $N$ are isogonal conjugates.
[i]Proof.[/i] Redefine $N$ as the isogonal conjugate of $M.$ It suffices to prove $N=(BQC)\cap(AQ).$ First, we claim $CBQN$ is cyclic. Indeed, $$\measuredangle BPC+\measuredangle BQC=\measuredangle BAC+180=\measuredangle BMC+\measuredangle BNC$$ and ... | [
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599,350 | Let $ABCD$ be a cyclic quadrilateral with center $O$.
Suppose the circumcircles of triangles $AOB$ and $COD$ meet again at $G$, while the circumcircles of triangles $AOD$ and $BOC$ meet again at $H$.
Let $\omega_1$ denote the circle passing through $G$ as well as the feet of the perpendiculars from $G$ to $AB$ and $CD$... | First, let $ E = AB \cap CD $ and $ F = BC \cap DA $ and denote the circumcircle of $ ABCD $ by $ \omega $. Let $ O_1, O_2 $ be the centers of $ \omega_1, \omega_2 $ respectively. Let $ M $ be the midpoint of $ GH $.
Consider the inversion about $ \omega $. It is clear that line $ AB $ goes to $ \omega_1 $ and t... | [
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599,351 | In triangle $ABC$ with incenter $I$ and circumcenter $O$, let $A',B',C'$ be the points of tangency of its circumcircle with its $A,B,C$-mixtilinear circles, respectively. Let $\omega_A$ be the circle through $A'$ that is tangent to $AI$ at $I$, and define $\omega_B, \omega_C$ similarly. Prove that $\omega_A,\omega_B,\... | I will fill in some intermediate steps in XmL's post.
Lemma 1: If $ X $, $ Y $ are the tangency points of the $ A $-mixtilinear incircle with $ AB, AC $ respectively then $ I $ is the midpoint of $ XY $.
[i]Proof:[/i] This is a well-known result, and moreover can be proved immediately with a limiting case of Saw... | [
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599,353 | We are given triangles $ABC$ and $DEF$ such that $D\in BC, E\in CA, F\in AB$, $AD\perp EF, BE\perp FD, CF\perp DE$. Let the circumcenter of $DEF$ be $O$, and let the circumcircle of $DEF$ intersect $BC,CA,AB$ again at $R,S,T$ respectively. Prove that the perpendiculars to $BC,CA,AB$ through $D,E,F$ respectively inters... | My solution:
Since the lines through $ A, B, C $ and perpendicular to $ EF, FD, DE $ are concurrent at the orthocenter of $\triangle DEF $ ,
so $ \triangle DEF $ and $ \triangle ABC $ are orthologic $ \Longrightarrow $ the perpendiculars through $ D, E, F $ to $ BC, CA, AB $ are concurrent at $ X $ .
Since $ AD, BE,... | [
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599,355 | "Define the Fibanocci sequence recursively by $F_1=1$, $F_2=1$ and $F_{i+2} = F_i + F_{i+1}$ for all(...TRUNCATED) | "Pretty easy. It is [i]well known[/i] that $F_i$ is strictly periodic mod $n$, to say that there exi(...TRUNCATED) | [-0.004518331494182348,0.0016061535570770502,-0.003071886021643877,0.005440608598291874,0.0238755419(...TRUNCATED) | [-0.008906040340662003,-0.033767297863960266,-0.004302870016545057,-0.013291704468429089,0.010613116(...TRUNCATED) | 2025-08-11 |
599,358 | "Let $d$ be a positive integer and let $\\varepsilon$ be any positive real. Prove that for all suffi(...TRUNCATED) | "I'm not sure whether my solution is correct or not, but I'll write it down anyway. \n\nLet $a$ be t(...TRUNCATED) | [-0.0021246045362204313,-0.020619377493858337,-0.0015944798942655325,0.021342255175113678,0.02896595(...TRUNCATED) | [0.038648203015327454,-0.02337401546537876,0.002447495935484767,0.012675275094807148,0.0364719294011(...TRUNCATED) | 2025-08-11 |
599,362 | "Given positive reals $a,b,c,p,q$ satisfying $abc=1$ and $p \\geq q$, prove that \\[ p \\left(a^2+b^(...TRUNCATED) | "Here is a slightly different solution: By AM-GM, we have $\\frac{1}{a} + \\frac{1}{b} + \\frac{1}{c(...TRUNCATED) | [0.021560685709118843,-0.02538418397307396,0.0031518267933279276,0.0334685742855072,-0.0266567487269(...TRUNCATED) | [0.03342369571328163,-0.01495265681296587,0.0014782147482037544,0.04623185470700264,-0.0286789163947(...TRUNCATED) | 2025-08-11 |
599,363 | "Let $a$, $b$, $c$ be positive reals. Prove that \\[ \\sqrt{\\frac{a^2(bc+a^2)}{b^2+c^2}}+\\sqrt{\\(...TRUNCATED) | "Another proof with expansion...\n\nBy Holder, \\[(\\text{LHS})^2\\left(\\sum a(b^2+c^2)\\right)\\le(...TRUNCATED) | [0.012603342533111572,-0.009711714461445808,0.0035407627001404762,0.06941498070955276,-0.02707886509(...TRUNCATED) | [0.00971190445125103,-0.01850649155676365,-0.0006880834116600454,0.05241920426487923,-0.039252668619(...TRUNCATED) | 2025-08-11 |
599,366 | "Let $ABC$ be a triangle with circumcenter $O$. Let $P$ be a point inside $ABC$, so let the points $(...TRUNCATED) | "[b]Generalization:[/b]\n\nLet $ D, E, F $ be the point on $ BC, CA, AB $ , respectively .\nLet $ K (...TRUNCATED) | [0.049597665667533875,-0.001226920518092811,-0.0035386502277106047,0.029201284050941467,0.0108298333(...TRUNCATED) | [0.05124698951840401,-0.0319947674870491,-0.002840761560946703,0.04044590890407562,0.004170460626482(...TRUNCATED) | 2025-08-11 |
599,368 | "Let $t$ and $n$ be fixed integers each at least $2$. Find the largest positive integer $m$ for whic(...TRUNCATED) | "We claim that $m=n$ is the desired maximum value. Note that the required condition is just a cute w(...TRUNCATED) | [-0.00140288844704628,-0.04768011346459389,-0.001136447419412434,0.022923873737454414,0.000554478378(...TRUNCATED) | [0.028700871393084526,-0.010161207057535648,-0.0048321629874408245,0.01022128015756607,0.01215499360(...TRUNCATED) | 2025-08-11 |
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