مساله3

In triangle ABC, angle B is a right angle. BD is the altitude. r1, and r2, are the inradii of the triangles ADB, and BDC respectively, the points E, F, and G are points of tangency. Prove that .

مساله2

Prove the following: if, in a triangle ABC,  mÐ BAO = mÐ OAC = 20°, mÐ ACO = 10°,and mÐ OCB = 30°, then mÐ OBC = 80°.

مساله

Prove the following: if, in a triangle ABC, median BD is such that mÐ A = mÐ DBC, and mÐ ADB = 45°, then mÐ A = 30°.

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(0,1)-Matrix	Four-Color Theorem	Prime Magic Square
Abhyankar's Conjecture	Gauss's Class Number C...	Prime Number Theorem
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Alternating Sign Matri...	Hadjicostas's Formula	Ramanujan's Square Equ...
Archimedes' Cattle Pro...	Heawood Conjecture	Reciprocal Fibonacci C...
Baudet's Conjecture	Hilbert's Problems	Refined Alternating Si...
Bellows Conjecture	Hofstadter-Conway $10,...	Robbins Conjecture
Bennequin's Conjecture	Honeycomb Conjecture	Robertson Conjecture
Bieberbach Conjecture	Jordan Curve Theorem	Ruth-Aaron Pair
Bimagic Square	Kaplan-Yorke Conjecture	Schmidt's Problem
Blaschke Conjecture	Kepler Conjecture	Serre's Conjecture
Burnside's Conjecture	Kneser's Conjecture	Serre's Problem
Busemann-Petty Problem	Lam's Problem	Shafarevich Conjecture
Carmichael's Conjecture	Macdonald's Constant-T...	Sierpinski's Conjecture
Catalan's Conjecture	Macdonald's Plane Part...	Smale's Problems
Central Binomial Coeff...	Magic Tour	Smith Conjecture
Dodecahedral Conjecture	Milin Conjecture	Snark
Double Bubble	Milnor's Conjecture	Solved Problems
Dyson's Conjecture	Mordell Conjecture	Stanley-Wilf Conjecture
Epsilon Conjecture	Netto's Conjecture	Strong Perfect Graph T...
Erdos-Heilbronn Conjec...	Nirenberg's Conjecture	Taniyama-Shimura Conje...
Erdos Squarefree Conje...	Perfect Graph Theorem	Tau Conjecture
Eternity	Perfect Magic Cube	Weisstein's Conjecture
Fermat's Last Theorem	Petersson Conjecture	Yff Conjecture
Flyping Conjecture	Power Series	Zarankiewicz's Conjecture
Four-Color Problem	Prime Arithmetic Progr...	Znám's Problem

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A-Sequence	Graph Spectrum	Prime Gaps
abc Conjecture	Grimm's Conjecture	Prime Power Conjecture
Ablowitz-Ramani-Segur...	Guy's Conjecture	Projective Plane
Abundance	Hadamard Matrix	Quasiperfect Number
Agoh's Conjecture	Halting Problem	Quillen-Lichtenbaum Co...
Almost Perfect Number	Happy End Problem	Rabin-Miller Strong Ps...
Alon-Tarsi Conjecture	Harborth Graph	Ramanujan-Petersson Co...
Alternating Series	Hard Square Entropy Co...	Random Close Packing
Andrica's Conjecture	Hardy-Littlewood Conje...	Rational Distance Problem
Artin's Conjecture	Harmonic Series	Rational Number
Baillie-PSW Primality...	Hasse's Conjecture	Riemann Hypothesis
Ball Triangle Picking	Heilbronn Triangle Pro...	Robin's Theorem
Barker Code	Hilbert's Problems	Ruth-Aaron Pair
Barnette's Conjecture	Hodge Conjecture	Sausage Conjecture
Baxter-Hickerson Function	Holyhedron	Schanuel's Conjecture
Beal's Conjecture	Hyperbolic Octahedron	Schinzel's Hypothesis
Beraha Constants	Hypercube Line Picking	Scholz Conjecture
Berry Conjecture	Idoneal Number	Selfridge's Conjecture
Borwein Conjectures	Isomorphic Factorization	Seymour Conjecture
Braun's Conjecture	Jacobian Conjecture	Shanks' Conjecture
Brocard's Conjecture	k-Tuple Conjecture	Shephard's Conjecture
Carmichael's Totient F...	Kaplan-Yorke Conjecture	Sierpinski's Composite...
Catalan's Aliquot Sequ...	Khinchin's Constant	Sierpinski Number of t...
Central Binomial Coeff...	Kimberling Sequence	Simon's Problems
Central Trinomial Coef...	Kobon Triangle	Simplex Simplex Picking
Class Number	Kreisel Conjecture	Smale's Problems
Coin Problem	Landau's Problems	Smarandache Function
Coin Tossing	Langlands Program	Social Golfer Problem
Collatz Problem	Langton's Ant	Solitary Number
Complementary Subspace...	Latin Square	Sophie Germain Prime
Completely Positive Ma...	Lehmer's Constant	Sphere Packing
Concurrent Normals Con...	Lehmer's Mahler Measur...	Sphere Tetrahedron Pic...
Constant Problem	Lehmer's Totient Problem	Spherical Code
Cramér Conjecture	Levy's Conjecture	Square Point Picking
Cramér-Granville Conje...	Local Density Conjecture	Strong Pseudoprime
Cube 3-Compound	Magic Square	Swinnerton-Dyer Conjec...
Cube Line Picking	Mandelbrot Set	Szpiro's Conjecture
Cube Triangle Picking	Mastermind	Tait's Knot Conjectures
Cyclic Number	Matchstick Graph	Tanc Function
de Polignac's Conjecture	Mersenne Prime	Taylor's Condition
Disk Triangle Picking	Mertens Theorem	Tetrahedral Number
Double Mersenne Number	Montgomery-Odlyzko Law	Tetrahedron Circumscri...
Earth-Moon Problem	Montgomery's Pair Corr...	Tetrahedron Line Picking
Eberhart's Conjecture	Mrs. Perkins's Quilt	Tetrahedron Triangle P...
Erdos-Ivic Conjecture	Multiplicative Persist...	Thurston Elliptization...
Erdos-Straus Conjecture	Net	Thurston's Geometrizat...
Erdos-Turán Conjecture	Newman's Conjecture	Totient Function
Eternity	Normal Number	Totient Valence Function
Euclid-Mullin Sequence	NP-Problem	Tower of Hanoi
Euler's Conjecture	Octahedral Number	Traveling Salesman Pro...
Euler-Mascheroni Constant	Octahedron Tetrahedron...	Triangle Triangle Picking
Euler Square	Odd Perfect Number	Trinomial Coefficient
Euler's Sum of Powers...	Orchard-Planting Problem	Twin Prime Conjecture
Exponential Integral	Ore's Conjecture	Twin Primes
Factorial Sums	P Versus NP Problem	Ulam's Conjecture
Fermat-Catalan Conjecture	Padé Conjecture	Unfolding
Fortunate Prime	Palindromic Number Con...	Uniformity Conjecture
Four Exponentials Conj...	Pentagonal Square Tria...	Union-Closed Sets Conj...
Friendly Number	Percolation Threshold	Unistable Polyhedron
Friendly Pair	Perfect Cuboid	Universal Turing Machine
Fuglede's Conjecture	Perfect Magic Cube	Unsolved Problems
Full Reptend Prime	Perfect Number	Vizing Conjecture
Gelfand's Question	Pi Formulas	Vojta's Conjecture
Generalized Moore Graph	Pillai's Conjecture	Wagstaff's Conjecture
Generalized Riemann Hy...	Poincaré Conjecture	Wall-Sun-Sun Prime
Gilbreath's Conjecture	Pollock's Conjecture	Waring's Prime Number...
Giuga's Conjecture	Pósa's Conjecture	Weird Number
Goldbach Conjecture	Positive Eigenvalued M...	Wilson Prime
Goldbach Partition	Powerful Number	Wolstenholme Prime
Golomb Ruler	Prime Array	Znám's Problem

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