2025 - Turkmenistan National Math Olympiad

by A_E_R, Apr 6, 2025, 9:48 AM

Let k,m,n>=2 positive integers and GCD(m,n)=1, Prove that the equation has infinitely many solutions in distict positive integers: x_1^m+x_2^m+⋯x_k^m=x_(k+1)^n

This post has been edited 1 time. Last edited by A_E_R, 3 hours ago
Reason: More understandable

turkmenistan

algebra

infinitely many solutions

Geometry

by Captainscrubz, Apr 6, 2025, 9:07 AM

Let $D$ be any point on side $BC$ of $\triangle ABC$ .Let $E$ and $F$ be points on $AB$ and $AC$ such that $EB=ED$ and $FD=FC$ respectively. Prove that the locus of circumcenter of $(DEF)$ is a line.
Prove without using moving points

This post has been edited 1 time. Last edited by Captainscrubz, 4 hours ago

Circumcenter

geometry

circumcircle

Find the constant

by JK1603JK, Apr 6, 2025, 7:21 AM

Find all $k$ such that $\left(a^{3}+b^{3}+c^{3}-3abc\right)^{2}-\left[a^{3}+b^{3}+c^{3}+3abc-ab(a+b)-bc(b+c)-ca(c+a)\right]^{2}\ge 2k\cdot(a-b)^{2}(b-c)^{2}(c-a)^{2}$ forall $a,b,c\ge 0.$

inequalities

P2 Geo that most of contestants died

by AlephG_64, Apr 5, 2025, 1:23 PM

Let $ABCD$ be a quadrilateral such that $\angle A$ and $\angle D$ are acute and $\overline{AB} = \overline{BC} = \overline{CD}$ . Suppose that $\angle BDA = 30^\circ$ , prove that $\angle DAC= 30^\circ$ .

This post has been edited 1 time. Last edited by AlephG_64, Yesterday at 1:23 PM

geometry

Vector geometry with unusual points

by Ciobi_, Apr 2, 2025, 12:28 PM

Let $\triangle ABC$ be an acute-angled triangle, with circumcenter $O$ , circumradius $R$ and orthocenter $H$ . Let $A_1$ be a point on $BC$ such that $A_1H+A_1O=R$ . Define $B_1$ and $C_1$ similarly.
If $\overrightarrow{AA_1} + \overrightarrow{BB_1} + \overrightarrow{CC_1} = \overrightarrow{0}$ , prove that $\triangle ABC$ is equilateral.

geometry

circumcircle

vector geometry

hard problem

by Cobedangiu, Mar 27, 2025, 2:54 PM

problem

inequalities

The last nonzero digit of factorials

by Tintarn, Mar 17, 2025, 12:21 PM

For each integer $n \ge 2$ we consider the last digit different from zero in the decimal expansion of $n!$ . The infinite sequence of these digits starts with $2,6,4,2,2$ . Determine all digits which occur at least once in this sequence, and show that each of those digits occurs in fact infinitely often.

number theory

number theory proposed

factorial

Digits

Fridolin just can't get enough from jumping on the number line

by Tintarn, Mar 17, 2025, 12:20 PM

Fridolin the frog jumps on the number line: He starts at $0$ , then jumps in some order on each of the numbers $1,2,\dots,9$ exactly once and finally returns with his last jump to $0$ . Can the total distance he travelled with these $10$ jumps be a) $20$ , b) $25$ ?

number theory

number theory proposed

Parallel Lines and Q Point

by taptya17, Dec 13, 2024, 8:34 AM

Let $\Delta ABC$ be an acute angled scalene triangle with circumcircle $\omega$ . Let $O$ and $H$ be the circumcenter and orthocenter of $\Delta ABC,$ respectively. Let $E,F$ and $Q$ be points on segments $AB,AC$ and $\omega$ , respectively, such that
$\angle BHE=\angle CHF=\angle AQH=90^\circ.$ Prove that $OQ$ and $AH$ intersect on the circumcircle of $\Delta AEF$ .

Proposed by Antareep Nath

This post has been edited 2 times. Last edited by taptya17, Dec 17, 2024, 6:08 AM

geometry

comp. geo starting with a 90-75-15 triangle. <APB =<CPQ, <BQA =<CQP.

by parmenides51, Sep 20, 2024, 9:25 PM

Let ABC be a triangle with $\angle A = 90^o$ , $\angle B = 75^o$ , and $AB = 2$ . Points $P$ and $Q$ of the sides $AC$ and $BC$ respectively, are such that $\angle APB = \angle CPQ$ and $\angle BQA = \angle CQP$ . Calculate the lenght of $QA$ .

This post has been edited 2 times. Last edited by parmenides51, Sep 20, 2024, 9:37 PM

geometry

right triangle

Submit

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math_exploration

by A_E_R, Apr 6, 2025, 9:48 AM

by Captainscrubz, Apr 6, 2025, 9:07 AM

by JK1603JK, Apr 6, 2025, 7:21 AM

by AlephG_64, Apr 5, 2025, 1:23 PM

by Ciobi_, Apr 2, 2025, 12:28 PM

by Cobedangiu, Mar 27, 2025, 2:54 PM

by Tintarn, Mar 17, 2025, 12:21 PM

by Tintarn, Mar 17, 2025, 12:20 PM

by taptya17, Dec 13, 2024, 8:34 AM

by parmenides51, Sep 20, 2024, 9:25 PM