MathDB
Fractional Diophantine

Source: 2009 SDMO Middle School Problem 3

8/24/2016
Find all ordered pairs of positive integers (a,b)\left(a,b\right) such that 1a+ab+1ab=1.\frac{1}{a}+\frac{a}{b}+\frac{1}{ab}=1.
Minima and maxima of a subset

Source: 2009 SDMO Middle School Problem 4

8/24/2016
Sally randomly chooses three different numbers from the set {1,2,…,14}\left\{1,2,\ldots,14\right\}. What is the probability that the sum of her smallest number and her biggest number is at least 1515?
probability
Flip heads to win!

Source: 2016 SDMO Middle School Problem 3

8/25/2016
Gwen, Eli, and Kat take turns flipping a coin in their respective order. The first one to flip heads wins. What is the probability that Kat will win?
A farmer and his trees

Source: 2009 SDMO Middle School Problem 1

8/24/2016
A farmer buys a batch of trees, which he wishes to plant in a square grid. For example, if he had 2525 trees, then he could plant them as shown below.
[asy] size(3cm,0); dot((0,0)); dot((0,1)); dot((0,2)); dot((0,3)); dot((0,4)); dot((1,0)); dot((1,1)); dot((1,2)); dot((1,3)); dot((1,4)); dot((2,0)); dot((2,1)); dot((2,2)); dot((2,3)); dot((2,4)); dot((3,0)); dot((3,1)); dot((3,2)); dot((3,3)); dot((3,4)); dot((4,0)); dot((4,1)); dot((4,2)); dot((4,3)); dot((4,4)); [/asy]
However, the farmer finds that he cannot plant his trees in a square grid. If he had 2020 more trees, or if he had 3939 fewer trees, then he could plant his trees in a square grid. How many trees did the farmer buy?
Square geometry

Source: 2009 SDMO Middle School Problem 2

8/24/2016
Let ABCDABCD be a square, and let EE and FF be points on sides AB‾\overline{AB} and CD‾\overline{CD}, respectively, such that AE:EB=AF:FD=2:1AE:EB=AF:FD=2:1. Let GG be the intersection of AF‾\overline{AF} and DE‾\overline{DE}, and let HH be the intersection of BF‾\overline{BF} and CE‾\overline{CE}. Find the ratio of the area of quadrilateral EGFHEGFH to the area of square ABCDABCD.
[asy] size(5cm,0); draw((0,0)--(3,0)); draw((3,0)--(3,3)); draw((3,3)--(0,3)); draw((0,3)--(0,0)); draw((0,0)--(2,3)); draw((1,0)--(3,3)); draw((0,3)--(1,0)); draw((2,3)--(3,0)); label("AA",(0,3),NW); label("BB",(3,3),NE); label("CC",(3,0),SE); label("DD",(0,0),SW); label("EE",(2,3),N); label("FF",(1,0),S); label("GG",(0.66666667,1),E); label("HH",(2.33333333,2),W); [/asy]
ratiogeometry
Quadratic and a not-prime

Source: 2016 SDMO High School Problem 1/2016 SDMO Middle School Problem 5

8/25/2016
Suppose aa and bb are integers such that x2+ax+b+1=0x^2+ax+b+1=0 has 22 positive integer solutions. Show that a2+b2a^2+b^2 is not prime.
quadratics
Digits and squares

Source: 2009 SDMO Middle School Problem 5

8/24/2016
Let A=33⋯3A=33\cdots3, where AA contains 20092009 33s. Let B=11⋯1088⋯89B=11\cdots1088\cdots89, where BB contains 20082008 11s and 20082008 88s. Prove that A2=BA^2=B.
Circle area ratio

Source: 2016 SDMO Middle School Problem 2

8/25/2016
Let ABAB be a diameter of a circle and let CC be a point on ABAB with 2⋅AC=BC2\cdot AC=BC. Let DD and EE be points on the circle such that DC⊥ABDC\perp AB and DEDE is a second diameter. What is the ratio of the area of △DCE\triangle{DCE} to the area of △ABD\triangle{ABD}?
geometryratio
Cool stuff about AB and BA

Source: Science ON 2021 grade XI/2

3/16/2021
Consider A,B∈Mn(C)A,B\in\mathcal{M}_n(\mathbb{C}) for which there exist p,q∈Cp,q\in\mathbb{C} such that pAB−qBA=InpAB-qBA=I_n. Prove that either (AB−BA)n=On(AB-BA)^n=O_n or the fraction pq\frac{p}{q} is well-defined (q≠0q \neq 0) and it is a root of unity.
(Sergiu Novac)
linear algebra
Pyramid of spheres

Source: 2016 SDMO Middle School Problem 4

8/25/2016
There is an infinitely tall tetrahedral stack of spheres where each row of the tetrahedron consists of a triangular arrangement of spheres, as shown below. There is 11 sphere in the top row (which we will call row 00), 33 spheres in row 11, 66 spheres in row 22, 1010 spheres in row 33, etc. The top-most sphere in row 00 is assigned the number 11. We then assign each other sphere the sum of the number(s) assigned to the sphere(s) which touch it in the row directly above it. Find a simplified expression in terms of nn for the sum of the numbers assigned to each sphere from row 00 to row nn.
[asy] import three; import solids; size(8cm);
//currentprojection = perspective(1, 1, 10);
triple backright = (-2, 0, 0), backleft = (-1, -sqrt(3), 0), backup = (-1, -sqrt(3) / 3, 2 * sqrt(6) / 3);
draw(shift(2 * backleft) * surface(sphere(1,20)), white); //2 draw(shift(backleft + backright) * surface(sphere(1,20)), white); //2 draw(shift(2 * backright) * surface(sphere(1,20)), white); //3 draw(shift(backup + backleft) * surface(sphere(1,20)), white); draw(shift(backup + backright) * surface(sphere(1,20)), white); draw(shift(2 * backup) * surface(sphere(1,20)), white);
draw(shift(backleft) * surface(sphere(1,20)), white); draw(shift(backright) * surface(sphere(1,20)), white); draw(shift(backup) * surface(sphere(1,20)), white);
draw(surface(sphere(1,20)), white);
label("Row 0", 2 * backup, 15 * dir(20)); label("Row 1", backup, 25 * dir(20)); label("Row 2", O, 35 * dir(20));
dot(-backup); dot(-7 * backup / 8); dot(-6 * backup / 8);
dot((backleft - backup) + backleft * 2); dot(5 * (backleft - backup) / 4 + backleft * 2); dot(6 * (backleft - backup) / 4 + backleft * 2);
dot((backright - backup) + backright * 2); dot(5 * (backright - backup) / 4 + backright * 2); dot(6 * (backright - backup) / 4 + backright * 2); [/asy]
geometry3D geometrypyramidspheretetrahedron
Punctual increasing

Source: Science ON 2021 grade XI/1

3/16/2021
Consider a function f:R→Rf:\mathbb{R}\rightarrow \mathbb{R}. For x∈Rx\in \mathbb{R} we say that ff is increasing in xx if there exists ϵx>0\epsilon_x > 0 such that f(x)≥f(a)f(x)\geq{f(a)}, ∀a∈(x−ϵx,x)\forall a\in (x-\epsilon_x,x) and f(x)≤f(b)f(x)\leq f(b), ∀b∈(x,x+ϵx)\forall b\in (x,x+\epsilon_x). <spanclass=′latex−bold′>(a)</span><span class='latex-bold'>(a)</span> Prove that if ff is increasing in xx, ∀x∈R\forall x\in \mathbb{R} then ff is increasing over R\mathbb{R}. <spanclass=′latex−bold′>(b)</span><span class='latex-bold'>(b)</span> We say that ff is increasing to the left in xx if there exists ϵx>0\epsilon_x > 0 such that f(x)≥f(a)f(x)\geq f(a) , ∀a∈(x−ϵx,x) \forall a \in (x-\epsilon_x,x). Provide an example of a function f:[0,1]→Rf: [0,1]\rightarrow \mathbb{R} for which there exists an infinite set M⊂(0,1)M \subset (0,1) such that ff is increasing to the left in every point of MM, yet ff is increasing over no proper subinterval of [0,1][0,1].
functionanalysisreal analysis
Clover convolution

Source: 2016 SDMO Middle School Problem 1

8/25/2016
Let ♣(x)\clubsuit\left(x\right) denote the sum of the digits of the positive integer xx. For example, ♣(8)=8\clubsuit\left(8\right)=8 and ♣(123)=1+2+3=6\clubsuit\left(123\right)=1+2+3=6. For how many two-digit values of xx is ♣(♣(x))=3\clubsuit\left(\clubsuit\left(x\right)\right)=3?
real analysis
The derivative of a determinant

Source: Science ON 2021 grade XI/3

3/16/2021
<spanclass=′latex−bold′>(a)</span><span class='latex-bold'>(a)</span> Let a,b∈Ra,b \in \mathbb{R} and f,g:R→Rf,g :\mathbb{R}\rightarrow \mathbb{R} be differentiable functions. Consider the function h(x)=∣abxf(a)f(b)f(x)g(a)g(b)g(x)∣h(x)=\begin{vmatrix} a &b &x\\ f(a) &f(b) &f(x)\\ g(a) &g(b) &g(x)\\ \end{vmatrix} Prove that hh is differentiable and find h′h'. \\ \\ <spanclass=′latex−bold′>(b)</span><span class='latex-bold'>(b)</span> Let n∈Nn\in \mathbb{N}, n≥3n\geq 3, take n−1n-1 pairwise distinct real numbers a1<a2<⋯<an−1a_1<a_2<\dots <a_{n-1} with sum ∑i=1n−1ai=0\sum_{i=1}^{n-1}a_i = 0, and consider n−1n-1 functions f1,f2,...fn−1:R→Rf_1,f_2,...f_{n-1}:\mathbb{R}\rightarrow \mathbb{R}, each of them n−2n-2 times differentiable over R\mathbb{R}. Prove that there exists a∈(a1,an−1)a\in (a_1,a_{n-1}) and θ,θ1,...,θn−1∈R\theta, \theta_1,...,\theta_{n-1}\in \mathbb{R}, not all zero, such that ∑k=1n−1θkak=θa\sum_{k=1}^{n-1} \theta_k a_k=\theta a and, at the same time, ∑k=1n−1θkfi(ak)=θfi(n−2)(a)\sum_{k=1}^{n-1}\theta_kf_i(a_k)=\theta f_i^{(n-2)}(a) for all i∈{1,2...,n−1}i\in\{1,2...,n-1\} . \\ \\ (Sergiu Novac)
differentiable functionreal analysis
Extra long, yet cute with invertible Z matrices

Source: Science ON 2021 grade XI/4

3/16/2021
Denote SL2(Z)\textrm{SL}_2 (\mathbb{Z}) and SL3(Z)\textrm{SL}_3 (\mathbb{Z}) the sets of matrices with 22 rows and 22 columns, respectively with 33 rows and 33 columns, with integer entries and their determinant equal to 11. <spanclass=′latex−bold′>(a)</span><span class='latex-bold'>(a)</span> Let NN be a positive integer and let gg be a matrix with 33 rows and 33 columns, with rational entries. Suppose that for each positive divisor MM of NN there exists a rational number qMq_M, a positive divisor f(M)f (M) of NN and a matrix γM∈SL3(Z)\gamma_M \in \textrm{SL}_3 (\mathbb{Z}) such that g=qM(10001000f(M))γM(10001000M). g = q_M \left(\begin{array}{ccc} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & f (M) \end{array}\right) \gamma_M \left(\begin{array}{ccc} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & M^{} \end{array}\right) . Moreover, if q1=1q_1 = 1, prove that det⁡(g)=N\det (g) = N and gg has the following shape: g=(a11a12Na13a21a22Na23Na31Na32Na33), g = \left(\begin{array}{ccc} a_{11} & a_{12} & Na_{13}\\ a_{21} & a_{22} & Na_{23}\\ Na_{31} & Na_{32} & Na_{33} \end{array}\right), where aija_{ij} are all integers, i,j∈{1,2,3}.i, j \in \{ 1, 2, 3 \} .
<spanclass=′latex−bold′>(b)</span><span class='latex-bold'>(b)</span> Provide an example of a matrix gg with 22 rows and 22 columns which satisfies the following properties: ∙\bullet For each positive divisor MM of 66 there exists a rational number qMq_M, a positive divisor f(M)f (M) of 66 and a matrix γM∈SL2(Z)\gamma_M \in \textrm{SL}_2 (\mathbb{Z}) such that g=qM(100f(M))γM(100M) g = q_M \left(\begin{array}{cc} 1 & 0\\ 0 & f (M) \end{array}\right) \gamma_M \left(\begin{array}{cc} 1 & 0\\ 0 & M^{} \end{array}\right) and q1=1q_1 = 1. ∙\bullet gg does not have its determinant equal to 66 and is not of the shape g=(a226a236a326a33), g = \left(\begin{array}{cc} a_{22} & 6 a_{23}\\ 6 a_{32} & 6 a_{33} \end{array}\right), where aija_{ij} are all positive integers, i,j∈{2,3}i, j \in \{ 2, 3 \}.
(Radu Toma)
linear algebranumber theory
Permute chessboard's rows

Source: Science ON 2021 Juniors/4

3/14/2021
An n×nn\times n chessboard is given, where nn is an even positive integer. On every line, the unit squares are to be permuted, subject to the condition that the resulting table has to be symmetric with respect to its main diagonal (the diagonal from the top-left corner to the bottom-right corner). We say that a board is alternative if it has at least one pair of complementary lines (two lines are complementary if the unit squares on them which lie on the same column have distinct colours). Otherwise, we call the board nonalternative. For what values of nn do we always get from the n×nn\times n chessboard an alternative board?\\ \\ (Alexandru Petrescu and Andra Elena Mircea)
combinatoricsinductionChessboard
Divisibility with options

Source: Science ON 2021 Juniors/1

3/14/2021
Let a,p,q∈Z≥1a,p,q\in \mathbb{Z}_{\ge 1} be such that aa is a perfect square, a=pqa=pq and 2021 ∣ p3+q3+p2q+pq2.2021~|~p^3+q^3+p^2q+pq^2. Prove that 20212021 divides a\sqrt a.\\ \\ (Cosmin Gavrilă)
number theory
Ants must fall off the table

Source: Science ON 2021 Seniors/3

3/14/2021
Let m,n∈Z≥1m,n\in \mathbb{Z}_{\ge 1} and a rectangular board m×nm\times n sliced by parallel lines to the rectangle's sides into mnmn unit squares. At moment t=0t=0, there is an ant inside every square, positioned exactly in its centre, such that it is oriented towards one of the rectangle's sides. Every second, all the ants move exactly a unit following their current orientation; however, if two ants meet at the centre of a unit square, both of them turn 180o180^o around (the turn happens instantly, without any loss of time) and the next second they continue their motion following their new orientation. If two ants meet at the midpoint of a side of a unit square, they just continue moving, without changing their orientation.\\ \\ Prove that, after finitely many seconds, some ant must fall off the table.\\ \\ (Oliver Hayman)
combinatoricspigeonhole
Radical axis parallel to some main line

Source: Science ON 2021 Juniors/3

3/14/2021
Circles ω1\omega_1 and ω2\omega_2 are externally tangent to each other at PP. A random line ℓ\ell cuts ω1\omega_1 at AA and CC and ω2\omega_2 at BB and DD (points A,C,B,DA,C,B,D are in this order on ℓ\ell). Line APAP meets ω2\omega_2 again at EE and line BPBP meets ω1\omega_1 again at FF. Prove that the radical axis of circles (PCD)(PCD) and (PEF)(PEF) is parallel to ℓ\ell. \\ \\ (Vlad Robu)
geometrycirclesAngle Chasing
&quot;Functional equation&quot; on sequences

Source: Science ON 2021 Seniors/1

3/14/2021
Find all sequences of positive integers (an)n≥1(a_n)_{n\ge 1} which satisfy an+2(an+1−1)=an(an+1+1)a_{n+2}(a_{n+1}-1)=a_n(a_{n+1}+1) for all n∈Z≥1n\in \mathbb{Z}_{\ge 1}. (Bogdan Blaga)
algebranumber theoryDivisibility
Maximum and minmum

Source: Science ON 2021 Juniors/2

3/14/2021
a,b,ca,b,c are nonnegative integers that satisfy a2+b2+c2=3a^2+b^2+c^2=3. Find the minimum and maximum value the sum 11+a+b+11+b+c+11+c+a\frac{1}{1+a+b}+\frac{1}{1+b+c}+\frac{1}{1+c+a} may achieve and find all a,b,ca,b,c for which equality occurs.\\ \\ (Andrei Bâra)
Inequalityalgebra
What could go wrong with some divisibility?

Source: Science ON 2021 Seniors/2

3/14/2021
Find all pairs (p,q)(p,q) of prime numbers such that pq−4 ∣ qp−1.p^q-4~|~q^p-1. (Vlad Robu)
number theoryDivisibilityOrder
Raise the digit get a square

Source: Science ON 2021 grade V/3

3/8/2021
A nonnegative integer nn is said to be <spanclass=′latex−italic′>squarish</span><span class='latex-italic'>squarish</span> is it satisfies the following conditions: <spanclass=′latex−bold′>(i)</span><span class='latex-bold'>(i)</span> it contains no digits of 99; <spanclass=′latex−bold′>(ii)</span><span class='latex-bold'>(ii)</span> no matter how we increase exactly one of its digits with 11, the outcome is a square.
Find all squarish nonnegative integers.
<spanclass=′latex−italic′>(VladRobu)</span><span class='latex-italic'>(Vlad Robu)</span>
number theory
Lots of circles

Source: Science ON 2021 Seniors/4

3/14/2021
ABCDABCD is a cyclic convex quadrilateral whose diagonals meet at XX. The circle (AXD)(AXD) cuts CDCD again at VV and the circle (BXC)(BXC) cuts ABAB again at UU, such that DD lies strictly between CC and VV and BB lies strictly between AA and UU. Let P∈AB∩CDP\in AB\cap CD.\\ \\ If MM is the intersection point of the tangents to UU and VV at (UPV)(UPV) and TT is the second intersection of circles (UPV)(UPV) and (PAC)(PAC), prove that ∠PTM=90o\angle PTM=90^o.\\ \\ (Vlad Robu)
geometryAngle ChasingcirclesProjective
primes with divisibility

Source: Science ON 2021 grade V/1

3/8/2021
Consider the prime numbers p1,p2,…,p2021p_1,p_2,\dots ,p_{2021} such that the sum p14+p24+⋯+p20214p_1^4+p_2^4+\dots +p_{2021}^4 is divisible by 60606060. Prove that at least 44 of these prime numbers are less than 20212021.
<spanclass=′latex−italic′>StefanBa˘la˘uca˘</span><span class='latex-italic'>Stefan Bălăucă</span>
number theory
Football championship

Source: Science ON 2021 grade V/2

3/8/2021
There is a football championship with 66 teams involved, such that for any 22 teams AA and BB, AA plays with BB and BB plays with AA (22 such games are distinct). After every match, the winning teams gains 33 points, the loosing team gains 00 points and if there is a draw, both teams gain 11 point each.\\ \\ In the end, the team standing on the last place has 1212 points and there are no 22 teams that scored the same amount of points.\\ \\ For all the remaining teams, find their final scores and provide an example with the outcomes of all matches for at least one of the possible final situations.
<spanclass=′latex−italic′>(AndreiBa^ra)</span><span class='latex-italic'>(Andrei Bâra)</span>
combinatorics