MathDB
The least monodigit multiple of 13

Source: Science ON 2021 grade V/4

3/8/2021
Find the least positive integer which is a multiple of 1313 and all its digits are the same.
(Adapted from Gazeta Matematică 1/1982, Florin Nicolăită)
number theory
geometry with 36 angles and lengths

Source: Science ON 2021 grade VI/2

3/8/2021
Is it possible for an isosceles triangle with all its sides of positive integer lengths to have an angle of 36o36^o?
(Adapted from Archimedes 2011, Traian Preda)
geometry
Some kind of pigeonhole principle

Source: Science ON 2021 grade VI/3

3/8/2021
Consider positive integers a<ba<b and the set C⊂{a,a+1,a+2,…,b−2,b−1,b}C\subset\{a,a+1,a+2,\dots ,b-2,b-1,b\}. Suppose CC has more than b−a+12\frac{b-a+1}{2} elements. Prove that there are two elements x,y∈Cx,y\in C that satisfy x+y=a+bx+y=a+b.
(From "Radu Păun" contest, Radu Miculescu)
Setspiepigeonhole principlecombinatorics
Find the invariant!

Source: Science ON 2021 grade VI/4

3/8/2021
The numbers 32\frac 32, 43\frac 43 and 65\frac 65 are intially written on the blackboard. A move consists of erasing one of the numbers from the blackboard, call it aa, and replacing it with bc−b−c+2bc-b-c+2, where b,cb,c are the other two numbers currently written on the blackboard. Is it possible that 1000999\frac{1000}{999} would eventually appear on the blackboard? What about 113108\frac{113}{108}?
(Andrei Bâra)
combinatoricsInvariantsinvariant
Curious conditions on a point

Source: Science ON 2021 grade VI/1

3/8/2021
Triangle ABCABC is such that ∠BAC>∠ABC>60o\angle BAC>\angle ABC>60^o. The perpendicular bisector of AB‾\overline{AB} intersects the segment BC‾\overline {BC} at OO. Suppose there exists a point DD on the segment AC‾\overline{AC} such that OD=ABOD=AB and ∠ODA=30o\angle ODA=30^o. Find ∠BAC\angle BAC.
(Vlad Robu)
geometry
Odd number generates all the integers

Source: Science ON 2021 grade VII/1

3/8/2021
Supoose AA is a set of integers which contains all integers that can be written as 2a−2b2^a-2^b, a,b∈Z≥1a,b\in \mathbb{Z}_{\ge 1} and also has the property that a+b∈Aa+b\in A whenever a,b∈Aa,b\in A. Prove that if AA contains at least an odd number, then A=ZA=\mathbb{Z}.
(Andrei Bâra)
Setsnumber theory
Isosceles triangle with angles of 44

Source: Science ON 2021 grade VII/2

3/8/2021
In triangle ABCABC, we have ∠ABC=∠ACB=44o\angle ABC=\angle ACB=44^o. Point MM is in its interior such that ∠MBC=16o\angle MBC=16^o and ∠MCB=30o\angle MCB=30^o. Prove that ∠MAC=∠MBC\angle MAC=\angle MBC.
(Andra Elena Mircea)
geometryangles
Can the Newton sums have such values over R?

Source: Science ON 2021 grade VII/3

3/8/2021
Are there any real numbers a,b,ca,b,c such that a+b+c=6a+b+c=6, ab+bc+ca=9ab+bc+ca=9 and a4+b4+c4=260a^4+b^4+c^4=260? What about if we let a4+b4+c4=210a^4+b^4+c^4=210?
(Andrei Bâra)
algebraInequalitysymmetric sumsNewton Sums
The gcd of 3 numbers instead of 2

Source: Science ON 2021 grade VIII/2

3/8/2021
Let n≥3n\ge 3 be an integer. Let s(n)s(n) be the number of (ordered) pairs (a;b)(a;b) consisting of positive integers a,ba,b from the set {1,2,…,n}\{1,2,\dots ,n\} which satisfy gcd⁡(a,b,n)=1\gcd (a,b,n)=1. Prove that s(n)s(n) is divisible by 44 for all n≥3n\ge 3.
(Vlad Robu)
combinatoricscounting
Find the right way to work with modulo

Source: Science ON 2021 grade VIII/1

3/8/2021
Are there any integers a,ba,b and cc, not all of them 00, such that a2=2021b2+2022c2  ?a^2=2021b^2+2022c^2~~?
(Cosmin Gavrilă)
number theorymodular arithmeticabstract algebra
Many terms, unusual inequality

Source: Science ON 2021 grade VIII/4

3/8/2021
Consider positive real numbers x,y,zx,y,z. Prove the inequality 1x+1y+1z+9x+y+z≥3((12x+y+1x+2y)+(12y+z+1y+2z)+(12z+x+1x+2z)).\frac 1x+\frac 1y+\frac 1z+\frac{9}{x+y+z}\ge 3\left (\left (\frac{1}{2x+y}+\frac{1}{x+2y}\right )+\left (\frac{1}{2y+z}+\frac{1}{y+2z}\right )+\left (\frac{1}{2z+x}+\frac{1}{x+2z}\right )\right ).
(Vlad Robu \& Sergiu Novac)
Inequalityinequalitiesalgebra
If it passes through the baricentre, it has some cool properties!

Source: Science ON 2021 grade VIII/3

3/8/2021
ABCDABCD is a scalene tetrahedron and let GG be its baricentre. A plane α\alpha passes through GG such that it intersects neither the interior of ΔBCD\Delta BCD nor its perimeter. Prove that dist(A,α)=dist(B,α)+dist(C,α)+dist(D,α).\textnormal{dist}(A,\alpha)=\textnormal{dist}(B,\alpha)+\textnormal{dist}(C,\alpha)+\textnormal{dist}(D,\alpha).
(Adapted from folklore)
geometry3D geometry
Sum of many sets

Source: Science ON 2021 grade VII/4

3/8/2021
Take k∈Z≥1k\in \mathbb{Z}_{\ge 1} and the sets A1,A2,…,AkA_1,A_2,\dots, A_k consisting of x1,x2,…,xkx_1,x_2,\dots ,x_k positive integers, respectively. For any two sets AA and BB, define A+B={a+b ∣ a∈A, b∈B}A+B=\{a+b~|~a\in A,~b\in B\}.
Find the least and greatest number of elements the set A1+A2+⋯+AkA_1+A_2+\dots +A_k may have.
(Andrei Bâra)
Setsequality casecombinatorics
Understanding how some sequence behaves

Source: Science ON 2021 grade IX/1

3/8/2021
Consider the sequence (an)n≥1(a_n)_{n\ge 1} such that a1=1a_1=1 and an+1=an+n2a_{n+1}=\sqrt{a_n+n^2}, ∀n≥1\forall n\ge 1. <spanclass=′latex−bold′>(a)</span><span class='latex-bold'>(a)</span> Prove that there is exactly one rational number among the numbers a1,a2,a3,…a_1,a_2,a_3,\dots. <spanclass=′latex−bold′>(b)</span><span class='latex-bold'>(b)</span> Consider the sequence (Sn)n≥1(S_n)_{n\ge 1} such that Sn=∑i=1n4(⌊ai+12⌋−⌊ai2⌋)(⌊ai+22⌋−⌊ai+12⌋).S_n=\sum_{i=1}^n\frac{4}{\left (\left \lfloor a_{i+1}^2\right \rfloor-\left \lfloor a_i^2\right \rfloor\right)\left(\left \lfloor a_{i+2}^2\right \rfloor-\left \lfloor a_{i+1}^2\right \rfloor\right)}. Prove that there exists an integer NN such that Sn>0.9S_n>0.9, ∀n>N\forall n>N.
(Stefan Obadă)
Sequencealgebra
Around O and H

Source: Science ON 2021 grade IX/2

3/8/2021
Consider the acute-angled triangle ABCABC, with orthocentre HH and circumcentre OO. DD is the intersection point of lines AHAH and BCBC and EE lies on AH‾\overline{AH} such that AE=DHAE=DH. Suppose EOEO and BCBC meet at FF. Prove that BD=CFBD=CF.
(Călin Pop & Vlad Robu)
geometry
Understanding the condition

Source: Science ON 2021 grade IX/3

3/8/2021
Real numbers a,b,ca,b,c with 0≤a,b,c≤10\le a,b,c\le 1 satisfy the condition a+b+c=1+2(1−a)(1−b)(1−c).a+b+c=1+\sqrt{2(1-a)(1-b)(1-c)}. Prove that 1−a2+1−b2+1−c2≤332.\sqrt{1-a^2}+\sqrt{1-b^2}+\sqrt{1-c^2}\le \frac{3\sqrt 3}{2}.
(Nora Gavrea)
Inequalityalgebraidentities
What quadrilaterals have all these conditions?

Source: Science ON 2021 grade IX/4

3/8/2021
<spanclass=′latex−bold′>(a)</span><span class='latex-bold'>(a)</span> On the sides of triangle ABCABC we consider the points M∈BC‾M\in \overline{BC}, N∈AC‾N\in \overline{AC} and P∈AB‾P\in \overline{AB} such that the quadrilateral MNAPMNAP with right angles ∠MNA\angle MNA and ∠MPA\angle MPA has an inscribed circle. Prove that MNAPMNAP has to be a kite. <spanclass=′latex−bold′>(b)</span><span class='latex-bold'>(b)</span> Is it possible for an isosceles trapezoid to be orthodiagonal and circumscribed too?
(Călin Udrea)
geometry
Aproximation for the power of a certain real number

Source: Science ON 2021 grade X/3

3/8/2021
Consider a real number aa that satisfies a=(a−1)3a=(a-1)^3. Prove that there exists an integer NN that satisfies ∣a2021−N∣<2−1000.|a^{2021}-N|<2^{-1000}.
(Vlad Robu)
complex numbersNewton SumsVietaalgebra
Unusual functional equation

Source: Science ON 2021 grade X/4

3/8/2021
Find all functions f:Z≥1→R>0f:\mathbb{Z}_{\ge 1}\to \mathbb{R}_{>0} such that for all positive integers nn the following relation holds: ∑d∣nf(d)3=(∑d∣nf(d))2,\sum_{d|n} f(d)^3=\left (\sum_{d|n} f(d) \right )^2, where both sums are taken over the positive divisors of nn.
(Vlad Robu)
functional equationnumber theorymultiplicative functionsalgebra
Inequality involving complex

Source: Science ON 2021 grade X/1

3/8/2021
Consider the complex numbers x,y,zx,y,z such that ∣x∣=∣y∣=∣z∣=1|x|=|y|=|z|=1. Define the number a=(1+xy)(1+yz)(1+zx).a=\left (1+\frac xy\right )\left (1+\frac yz\right )\left (1+\frac zx\right ). <spanclass=′latex−bold′>(a)</span><span class='latex-bold'>(a)</span> Prove that aa is a real number. <spanclass=′latex−bold′>(b)</span><span class='latex-bold'>(b)</span> Find the minimal and maximal value aa can achieve, when x,y,zx,y,z vary subject to ∣x∣=∣y∣=∣z∣=1|x|=|y|=|z|=1.
(Stefan Bălăucă & Vlad Robu)
algebracomplex numbersInequality
Cute characterization for Riemann-integrable functions

Source: Science ON 2021 grade XII/3

3/16/2021
Define E⊆{f:[0,1]→R∣f is Riemann-integrable}E\subseteq \{f:[0,1]\to \mathbb{R}\mid f \textnormal{ is Riemann-integrable}\} such that EE posseses the following properties:\\ <spanclass=′latex−bold′>(i)</span><span class='latex-bold'>(i)</span> If ∫01f(x)g(x)dx=0\int_0^1 f(x)g(x) dx = 0 for f∈Ef\in E with ∫01f2(t)dt≠0\int_0^1f^2(t)dt \neq 0, then g∈Eg\in E; \\ <spanclass=′latex−bold′>(ii)</span><span class='latex-bold'>(ii)</span> There exists h∈Eh\in E with ∫01h2(t)dt≠0\int_0^1 h^2(t)dt\neq 0.\\ Prove that E={f:[0,1]→R∣f is Riemann-integrable}E=\{f:[0,1]\to \mathbb{R}\mid f \textnormal{ is Riemann-integrable}\}. \\ (Andrei Bâra)
integrabilityriemann integrabilityfunctionreal analysis
All nontrivial elements have infinite order

Source: Science ON 2021 grade XII/4

3/16/2021
Consider a group GG with at least 22 elements and the property that each nontrivial element has infinite order. Let HH be a cyclic subgroup of GG such that the set {xH∣x∈G}\{xH\mid x\in G\} has 22 elements. \\ <spanclass=′latex−bold′>(a)</span><span class='latex-bold'>(a)</span> Prove that GG is cyclic. \\ <spanclass=′latex−bold′>(b)</span><span class='latex-bold'>(b)</span> Does the conclusion from <spanclass=′latex−bold′>(a)</span><span class='latex-bold'>(a)</span> stand true if GG contains nontrivial elements of finite order?
superior algebragroup theory
Find the functions!

Source: Science ON 2021 grade XII/1

3/16/2021
Find all differentiable functions f,g:[0,∞)→Rf, g:[0,\infty) \to \mathbb{R} and the real constant k≥0k\geq 0 such that \begin{align*} f(x) &=k+ \int_0^x \frac{g(t)}{f(t)}dt \\ g(x) &= -k-\int_0^x f(t)g(t) dt \end{align*} and f(0)=k,f′(0)=−k2/3f(0)=k, f'(0)=-k^2/3 and also f(x)≠0f(x)\neq 0 for all x≥0x\geq 0.\\ \\ (Nora Gavrea)
functional equationintegrationdifferentiable functioncalculus
Interesting condition on two elements of a certain ring

Source: Science ON 2021 grade XII/2

3/16/2021
Consider an odd prime pp. A comutative ring (A,+,⋅)(A,+, \cdot) has the property that ab=0ab=0 implies ap=0a^p=0 or bp=0b^p=0. Moreover, 1+1+⋯+1⏟p times=0\underbrace{1+1+\cdots +1}_{p \textnormal{ times}} =0. Take x,y∈Ax,y\in A such that there exist m,n≥1m,n\geq 1, m≠nm\neq n with x+y=xmy=xnyx+y=x^my=x^ny, and also yy is not invertible. \\ \\ <spanclass=′latex−bold′>(a)</span><span class='latex-bold'>(a)</span> Prove that (a+b)p=ap+bp(a+b)^p=a^p+b^p and (a+b)p2=ap2+bp2(a+b)^{p^2}=a^{p^2}+b^{p^2} for all a,b∈Aa,b\in A.\\ <spanclass=′latex−bold′>(b)</span><span class='latex-bold'>(b)</span> Prove that xx and yy are nilpotent.\\ <spanclass=′latex−bold′>(c)</span><span class='latex-bold'>(c)</span> If yy is invertible, does the conclusion that xx is nilpotent stand true? \\ \\ (Bogdan Blaga)
Ringsabstract algebrasuperior algebra
Sticks forming non-degenerate triangles

Source: Kyiv City MO 2022 Round 1, Problem 7.2

1/23/2022
There are nn sticks which have distinct integer length. Suppose that it's possible to form a non-degenerate triangle from any 33 distinct sticks among them. It's also known that there are sticks of lengths 55 and 1212 among them. What's the largest possible value of nn under such conditions?
(Proposed by Bogdan Rublov)
combinatoricsgeometry