Problem Solving | Beni Bogoşel's blog

IMO 2023 Problem 1

July 12, 2023 beni22sof Leave a comment

Problem 1. Determine all composite integers ${n>1}$ that satisfy the following property: if ${d_1, d_2, \ldots, d_k}$ are all the positive divisors of ${n}$ with ${1=d_1<d_2<\cdots<d_k=n}$ , then ${d_i}$ divides ${d_{i+1}+d_{i+2}}$ for every ${1 \leqslant i \leqslant k-2}$ .

Problems of the International Mathematical Olympiad 2023

July 11, 2023 beni22sof Leave a comment

Problem 2. Let ${ABC}$ be an acute-angled triangle with ${AB < AC}$ . Let ${\Omega}$ be the circumcircle of ${ABC}$ . Let ${S}$ be the midpoint of the arc ${CB}$ of ${\Omega}$ containing ${A}$ . The perpendicular from ${A}$ to ${BC}$ meets ${BS}$ at ${D}$ and meets ${\Omega}$ again at ${E \neq A}$ . The line through ${D}$ parallel to ${BC}$ meets line ${BE}$ at ${L}$ . Denote the circumcircle of triangle ${BDL}$ by ${\omega}$ . Let ${\omega}$ meet ${\Omega}$ again at ${P \neq B}$ . Prove that the line tangent to ${\omega}$ at ${P}$ meets line ${BS}$ on the internal angle bisector of ${\angle BAC}$ .

Problem 3. For each integer ${k \geqslant 2}$ , determine all infinite sequences of positive integers ${a_1, a_2, \ldots}$ for which there exists a polynomial ${P}$ of the form ${P(x)=x^k+c_{k-1} x^{k-1}+\cdots+c_1 x+c_0}$ , where ${c_0, c_1, \ldots, c_{k-1}}$ are non-negative integers, such that

$\displaystyle P\left(a_n\right)=a_{n+1} a_{n+2} \cdots a_{n+k}$

for every integer ${n \geqslant 1}$ .

Problem 4. Let ${x_1,x_2,\dots,x_{2023}}$ be pairwise different positive real numbers such that

$\displaystyle a_n=\sqrt{(x_1+x_2+\dots+x_n)\left(\frac{1}{x_1}+\frac{1}{x_2}+\dots+\frac{1}{x_n}\right)}$

is an integer for every ${n=1,2,\dots,2023.}$ Prove that ${a_{2023} \geqslant 3034.}$

Problem 5. Let ${n}$ be a positive integer. A Japanese triangle consists of ${1 + 2 + \dots + n}$ circles arranged in an equilateral triangular shape such that for each ${i = 1}$ , ${2}$ , ${\dots}$ , ${n}$ , the ${i^{th}}$ row contains exactly ${i}$ circles, exactly one of which is coloured red. A ninja path in a Japanese triangle is a sequence of ${n}$ circles obtained by starting in the top row, then repeatedly going from a circle to one of the two circles immediately below it and finishing in the bottom row. Here is an example of a Japanese triangle with ${n = 6}$ , along with a ninja path in that triangle containing two red circles.

In terms of ${n}$ , find the greatest ${k}$ such that in each Japanese triangle there is a ninja path containing at least ${k}$ red circles.

Problem 6. Let ${ABC}$ be an equilateral triangle. Let ${A_1,B_1,C_1}$ be interior points of ${ABC}$ such that ${BA_1=A_1C}$ , ${CB_1=B_1A}$ , ${AC_1=C_1B}$ , and

$\displaystyle \angle BA_1C+\angle CB_1A+\angle AC_1B=480^\circ$

Let ${BC_1}$ and ${CB_1}$ meet at ${A_2,}$ let ${CA_1}$ and ${AC_1}$ meet at ${B_2,}$ and let ${AB_1}$ and ${BA_1}$ meet at ${C_2.}$ Prove that if triangle ${A_1B_1C_1}$ is scalene, then the three circumcircles of triangles ${AA_1A_2, BB_1B_2}$ and ${CC_1C_2}$ all pass through two common points.

(Note: a scalene triangle is one where no two sides have equal length.)

Source: imo-official.org, AOPS forums

Categories: Algebra, Combinatorics, Geometry, IMO, Inequalities, Number theory, Olympiad, Problem Solving Tags: Geometry, IMO, number theory, Problem Solving

Applications of Helly’s theorem

March 12, 2023 beni22sof Leave a comment

Prove that if the plane can be covered with $n \geq 3$ half planes then there exist three of these which also cover the plane.
On a circle consider a finite set of arcs which do not cover the circle, such that any two of them have non-void intersection. Show that all arcs have a common points. If the arcs cover the circle does the conclusion still hold?
Consider $n \geq 3$ half circles which cover the whole circle. Show that we can pick three of them which still cover the circle.

I’ll not provide the solutions for now. The title should be a strong indication towards finding a solution!

Categories: Combinatorics, Geometry, Olympiad Tags: convex, Geometry, helly, Problem Solving

IMO 2022 – Problem 4 – Geometry

September 6, 2022 beni22sof Leave a comment

Problem 4. Let ${ABCDE}$ be a convex pentagon such that ${BC = DE}$ . Assume that there is a point ${T}$ inside ${ABCDE}$ with ${TB = TD}$ , ${TC = TE}$ and ${\angle ABT = \angle TEA}$ . Let line ${AB}$ intersect lines ${CD}$ and ${CT}$ at points ${P}$ and ${Q}$ , respectively. Assume that the points ${P, B, A, Q}$ occur on their line in that order. Let line ${AE}$ intersect lines ${CD}$ and ${DT}$ at points ${R}$ and ${S}$ , respectively. Assume that the points ${R, E, A, S}$ occur on their line in that order. Prove that the points ${P, S, Q, R}$ lie on a circle.

Balkan Mathematical Olympiad 2022

May 9, 2022 beni22sof Leave a comment

Problem 1. Let ${ABC}$ be an acute triangle such that ${CA \neq CB}$ with circumcircle ${\omega}$ and circumcentre ${O}$ . Let ${t_A}$ and ${t_B}$ be the tangents to ${\omega}$ at ${A}$ and ${B}$ respectively, which meet at ${X}$ . Let ${Y}$ be the foot of the perpendicular from ${O}$ onto the line segment ${CX}$ . The line through ${C}$ parallel to line ${AB}$ meets ${t_A}$ at ${Z}$ . Prove that the line ${YZ}$ passes through the midpoint of the line segment ${AC}$ .

Problem 2. Let ${a, b}$ and ${n}$ be positive integers with ${a>b}$ such that all of the following hold:

i. ${a^{2021}}$ divides ${n}$ ,

ii. ${b^{2021}}$ divides ${n}$ ,

iii. 2022 divides ${a-b}$ .

Prove that there is a subset ${T}$ of the set of positive divisors of the number ${n}$ such that the sum of the elements of ${T}$ is divisible by 2022 but not divisible by ${2022^2}$ .

Problem 3. Find all functions ${f: (0, \infty) \rightarrow (0, \infty)}$ such that $f(y(f(x))^3 + x) = x^3f(y) + f(x)$ for all ${x, y>0}$ .

Problem 4. Consider an ${n \times n}$ grid consisting of ${n^2}$ until cells, where ${n \geq 3}$ is a given odd positive integer. First, Dionysus colours each cell either red or blue. It is known that a frog can hop from one cell to another if and only if these cells have the same colour and share at least one vertex. Then, Xanthias views the colouring and next places ${k}$ frogs on the cells so that each of the ${n^2}$ cells can be reached by a frog in a finite number (possible zero) of hops. Find the least value of ${k}$ for which this is always possible regardless of the colouring chosen by Dionysus.

Source: AOPS

Categories: Algebra, Combinatorics, Geometry, IMO, Olympiad, Problem Solving Tags: balkan, IMO, Problem Solving

Putnam 2021 – Day 1

February 7, 2022 beni22sof Leave a comment

Here you can find the problems of the first day of the Putnam 2021 contest. Source.

A1. A grasshopper starts at the origin in the coordinate plane and makes a sequence of hops. Each hop has length ${5}$ , and after each hop the grasshopper is at a point whose coordinates are both integers; thus, there are ${12}$ possible locations for the grasshopper after the first hop. What is the smallest number of hops needed for the grasshopper to reach the point ${(2021,2021)}$ ?

A2. For every positive real number ${x}$ , let

$\displaystyle g(x) = \lim_{r\rightarrow 0} \left((x+1)^{r+1}-x^{r+1}\right)^{\frac{1}{r}}.$

Find ${\lim_{x \rightarrow \infty} \frac{g(x)}{x}}$ .

A3. Determine all positive integers ${N}$ for which the sphere

$\displaystyle x^2+y^2+z^2=N$

has an inscribed regular tetrahedron whose vertices have integer coordinates.

A4. Let

$\displaystyle I(R)= \iint_{x^2+y^2\leq R^2} \left( \frac{1+2x^2}{1+x^4+6x^2y^2+y^4}-\frac{1+y^2}{2+x^4+y^4}\right) dx dy.$

Find ${\lim_{R\rightarrow \infty} I(R)}$ , or show that this limit does not exist.

A5. Let ${A}$ be the set of all integers ${n}$ such that ${1 \leq n \leq 2021}$ and ${\gcd (n,2021)=1}$ . For every nonnegative integer ${j}$ , let

$\displaystyle S_j = \sum_{n\in A} n^j.$

Determine all values of ${j}$ such that ${S(j)}$ is a multiple of ${2021}$ .

A6. Let ${P(x)}$ be a polynomial whose coefficients are all either ${0}$ or ${1}$ . Suppose that ${P(x)}$ can be written as a product of two nonconstant polynomials with integer coefficients. Does it follow that ${P(2)}$ is a composite integer?

Categories: Algebra, Analysis, IMO, Number theory, Olympiad, Problem Solving Tags: Analysis, olympiad, Problem Solving, putnam

Putnam 2020 problems – Day 1

March 19, 2021 beni22sof Leave a comment

A1. How many positive integers ${N}$ satisfy all of the following three conditions?

(i) ${N}$ is divisible by ${2020}$ .
(ii) ${N}$ has at most ${2020}$ decimal digits.
(iii) The decimal digits of ${N}$ are a string of consecutive ones followed by a string of consecutive zeros.

A2. Let ${k}$ be a nonnegative integer. Evaluate

$\displaystyle \sum_{j=0}^k 2^{k-j} {k+j \choose j}.$

A3. Let ${a_0=\pi/2}$ and ${a_n = \sin(a_{n-1})}$ for ${n\geq 1}$ . Determine whether

$\displaystyle \sum_{n=1}^\infty a_n^2$

converges.

A4. Consider a horizontal strip of ${N+2}$ squares in which the first and the last square are black and the remaining ${N}$ squares are all white. Choose a white square uniformly at random, choose one of its two neighbors with equal probability, and color this neighboring square black if it is not already black. Repeat this process untill all the remaining white squares have only black neighbors. Let ${w(N)}$ be the expected number of white squares remaining. Find

$\displaystyle \lim_{N \rightarrow \infty} \frac{w(N)}{N}.$

A5. Let ${a_n}$ be the number of sets ${S}$ of positive integers for which

$\displaystyle \sum_{ k \in S} F_k = n,$ where the Fibonacci sequence ${(F_k)_{k\geq 1}}$ satisfies ${F_{k+2}=F_{k+1}+F_k}$ and begins with ${F_1=1,F_2=1,F_3=2,F_4=3}$ . Find the largest integer ${n}$ such that ${a_n = 2020}$ .

A6. For a positive integer ${N}$ , let ${f_N}$ be the function defined by

$\displaystyle f_N(x) = \sum_{n=0}^N \frac{N+1/2-n}{(N+1)(2n+1)}\sin ((2n+1)x).$ Determine the smallest constant ${M}$ such that ${f_N(x)\leq M}$ for all ${N}$ and all real ${x}$ .

Source: https://kskedlaya.org/putnam-archive/

Categories: Algebra, Analysis, Olympiad, Probability, Problem Solving, Real Analysis Tags: Analysis, Problem Solving, putnam

IMC 2020 Day 1 – Some Hints for Problems 1-2

August 31, 2020 beni22sof 1 comment

Problem 1. Let ${n}$ be a positive integer. Compute the number of words ${w}$ (finite sequences of letters) that satisfy all the following requirements: (1) ${w}$ consists of ${n}$ letters, all of them from the alphabet ${\{a,b,c,d\}}$ (2) ${w}$ contains an even number of letters ${a}$ (3) ${w}$ contains an even number of letters ${b}$ (For example, for ${n=2}$ there are ${6}$ such words: ${aa, bb, cc,dd,cd}$ and ${dc}$ .)

(proposed by Armend Sh. Shabani, University of Prishtina)

Hint: In order to get a formula for the total number of words it is enough to note that the even number $2k$ of a’s can be distributed in ${n\choose 2k}$ ways among the possible positions, and the even number of $2l$ b’s can be distributed in ${n-2k\choose 2l}$ ways among the remaining positions. Once the a’s and b’s are there, the rest can be filled with c’s and d’s in $2^{n-2k-2l}$ ways. This gives

$\displaystyle \sum_{k=0}^{n/2} \sum_{l=0}^{n/2-k}{n\choose 2k}{n-2k\choose 2l}2^{n-2k-2l}$

Next, use some tricks regarding expansions of $(a+b)^n+(a-b)^n$ to compress the sum above to $4^{n-1}+2^{n-1}$ .

Problem 2. Let ${A}$ and ${B}$ be ${n\times n}$ real matrices such that

$\displaystyle \text{rank}(AB-BA+I) = 1,$

where ${I}$ is the ${n\times n}$ identity matrix. Prove that

$\displaystyle \text{tr}(ABAB)-\text{tr}(A^2B^2) = \frac{1}{2}n(n-1).$

(proposed by Rustam Turdibaev, V.I. Romanovskiy Institute of Mathematics)

Hint: Every time you hear about a rank 1 matrix you should think of “column matrix times line matrix”. Indeed, writing $AB-BA+I = cd^T$ with $c,d$ column vectors, gives $AB-BA = cd^T-I$ . Moreover, taking trace in the above equality and using the fact that you can perform circular permutations in products in traces, you obtain that $tr(cd^T) = d^Tc = n$ .

Moreover, squaring $AB-BA = cd^T-I$ gives

$\displaystyle ABAB+BABA-ABBA-BAAB = d^Tc(cd^T) - 2d^Tc+I$

Taking trace above gives the desired result!

Categories: Algebra, Combinatorics, Higher Algebra, IMO, Linear Algebra, Olympiad, Problem Solving Tags: Combinatorics, IMC, linear algebra, Problem Solving

IMC 2020 Online – Problems – Day 1

August 28, 2020 beni22sof Leave a comment

(proposed by Armend Sh. Shabani, University of Prishtina)

Problem 2. Let ${A}$ and ${B}$ be ${n\times n}$ real matrices such that

$\displaystyle \text{rank}(AB-BA+I) = 1,$

where ${I}$ is the ${n\times n}$ identity matrix. Prove that

$\displaystyle \text{tr}(ABAB)-\text{tr}(A^2B^2) = \frac{1}{2}n(n-1).$

(proposed by Rustam Turdibaev, V.I. Romanovskiy Institute of Mathematics)

Problem 3. Let ${d \geq 2}$ be an integer. Prove that there exists a constant ${C(d)}$ such that the following holds: For any convex polytope ${K\subset \Bbb{R}^d}$ , which is symmetric about the origin, and any ${\varepsilon \in (0,1)}$ , there exists a convex polytope ${L \subset \Bbb{R}^d}$ with at most ${C(d) \varepsilon^{1-d}}$ vertices such that

$\displaystyle (1-\varepsilon)K \subseteq L \subset K.$

(For a real ${\alpha}$ , a set ${T\subset \Bbb{R}^d}$ with nonempty interior is a convex polytope with at most ${\alpha}$ vertices if ${T}$ is a convex hull of a set ${X \subset \Bbb{R}^d}$ of at most ${\alpha}$ points.)

(proposed by Fedor Petrov, St. Petersburg State University)

Problem 4. A polynomial ${p}$ with real coefficients satisfies the equation ${p(x+1)-p(x) = x^{100}}$ for all ${x \in \Bbb{R}}$ . Prove that ${p(1-t)\geq p(t)}$ for ${0\leq t\leq 1/2}$ .

(proposed by Daniil Klyuev, St. Petersburg State University)

source: http://www.imc-math.org

Categories: Affine Geometry, Algebra, Combinatorics, Higher Algebra, IMO, Problem Solving Tags: IMC, olympiad, Problem Solving

IMC 2019 – Problems from Day 1

July 30, 2019 beni22sof Leave a comment

Problem 1. Evaluate the product

$\displaystyle \prod_{n=3}^\infty \frac{(n^3+3n)^2}{n^6-64}.$

Problem 2. A four-digit ${YEAR}$ is called very good if the system

$\displaystyle \left\{\begin{array}{rcl} Yx+Ey+Az+Rw & = & Y \\ Rx+Yy+Ez+Aw & = & E \\ Ax+Ry+Yz+Ew & = & A \\ Ex+Ay+Rz+Yw & = & R \end{array} \right.$

of linear equations in the variables ${x,y,z,w}$ has at least two solutions. Find all very good ${YEAR}$ s in the ${21}$ st century (between ${2001}$ and ${2100}$ ).

Problem 3. Let ${f : (-1,1) \rightarrow \Bbb{R}}$ be a twice differentiable function such that

$\displaystyle 2f'(x)+xf''(x)\geq 1 \text{ for } x \in (-1,1).$

Prove that

$\displaystyle \int_{-1}^1 xf(x) dx \geq \frac{1}{3}.$

Problem 4. Define the sequence ${a_0,a_1,...}$ of numbers by the following recurrence:

$\displaystyle a_0 = 1, \ \ a_1 = 2, \ \ (n+3)a_{n+2} = (6n+9)a_{n+1}-na_n \text{ for } n \geq 0.$

Prove that all terms of this sequence are integers.

Problem 5. Determine whether there exist an odd positive integer ${n}$ and ${n \times n}$ matrices ${A}$ and ${B}$ with integer entries that satisfy the following conditions:

${\det B = 1}$ .
${AB = BA}$ .
${A^4+4A^2B^2+16B^4 = 2019 I}$

As usual ${I}$ denotes the ${n\times n}$ identity matrix.

Source: imc-math.org.uk

Categories: Algebra, Analysis, Higher Algebra, IMO, Linear Algebra, Number theory, Olympiad, Problem Solving Tags: Algebra, Analysis, IMC, olympiad, Problem Solving

Putnam 2018 – Problem B2

February 7, 2019 beni22sof Leave a comment

B2. Let ${n}$ be a positive integer and let ${f_n(z) = n+(n-1)z+(n-2)z^2+...+z^{n-1}}$ . Prove that ${f_n}$ has no roots in the closed unit disk ${\{z \in \Bbb{C}: |z| \leq 1\}}$ .

Putnam 2018, Problem B2

Solution: This is a cute problem and the path to the solution is quite straightforward once you notice some obvious things. First note that ${f_n(0)=n}$ , so ${0}$ is not a solution.

IMO 2018 Problems – Day 1

July 9, 2018 beni22sof Leave a comment

Problem 1. Let ${\Gamma}$ be the circumcircle of acute triangle ${ABC}$ . Points ${D}$ and ${E}$ are on segments ${AB}$ and ${AC}$ respectively such that ${AD = AE}$ . The perpendicular bisectors of ${BD}$ and ${CE}$ intersect minor arcs ${AB}$ and ${AC}$ of ${\Gamma}$ at points ${F}$ and ${G}$ respectively. Prove that lines ${DE}$ and ${FG}$ are either parallel or they are the same line.

Problem 2. Find all integers ${n \geq 3}$ for which there exist real numbers ${a_1, a_2, \dots a_{n + 2}}$ satisfying ${a_{n + 1} = a_1}$ , ${a_{n + 2} = a_2}$ and

$\displaystyle a_ia_{i + 1} + 1 = a_{i + 2}$

For ${i = 1, 2, \dots, n}$ .

Problem 3. An anti-Pascal triangle is an equilateral triangular array of numbers such that, except for the numbers in the bottom row, each number is the absolute value of the difference of the two numbers immediately below it. For example, the following is an anti-Pascal triangle with four rows which contains every integer from ${1}$ to ${10}$

$\displaystyle 4$

$\displaystyle 2\quad 6$

$\displaystyle 5\quad 7 \quad 1$

$\displaystyle 8\quad 3 \quad 10 \quad 9$

Does there exist an anti-Pascal triangle with ${2018}$ rows which contains every integer from ${1}$ to ${1 + 2 + 3 + \dots + 2018}$ ?

Source: AoPS.

Categories: Geometry, IMO, Olympiad, Uncategorized Tags: Geometry, IMO, olympiad, Problem Solving

Balkan Mathematical Olympiad 2018

June 23, 2018 beni22sof Leave a comment

Problem 1. A quadrilateral ${ABCD}$ is inscribed in a circle ${k}$ , where ${AB>CD}$ and ${AB}$ is not parallel to ${CD}$ . Point ${M}$ is the intersection of the diagonals ${AC}$ and ${BD}$ and the perpendicular from ${M}$ to ${AB}$ intersects the segment ${AB}$ at the point ${E}$ . If ${EM}$ bisects the angle ${CED}$ , prove that ${AB}$ is a diameter of the circle ${k}$ .

Problem 2. Let ${q}$ be a positive rational number. Two ants are initially at the same point ${X}$ in the plane. In the ${n}$ -th minute ${(n=1,2,...)}$ each of them chooses whether to walk due north, east, south or west and then walks the distance of ${q^n}$ meters. After a whole number of minutes, they are at the same point in the plane (non necessarily ${X}$ ), but have not taken exactly the same route within that time. Determine all the possible values of ${q}$ .

Problem 3. Alice and Bob play the following game: They start with two non-empty piles of coins. Taking turns, with Alice playing first, each player chooses a pile with an even number of coins and moves half of the coins of this pile to the other pile. The came ends if a player cannot move, in which case the other player wins.

Determine all pairs ${(a,b)}$ of positive integers such that if initially the two piles have ${a}$ and ${b}$ coins, respectively, then Bob has a winning strategy.

Problem 4. Find all primes ${p}$ and ${q}$ such that ${3p^{q-1}+1}$ divides ${11^q+17^p}$ .

Source: https://bmo2018.dms.rs/wp-content/uploads/2018/05/BMOproblems2018_English.pdf

Categories: Algebra, Combinatorics, Geometry, IMO, Number theory, Olympiad Tags: BMO, Geometry, olympiad, Problem Solving

Romanian Masters in Mathematics contest – 2018

March 9, 2018 beni22sof Leave a comment

Problem 1. Let ${ABCD}$ be a cyclic quadrilateral an let ${P}$ be a point on the side ${AB.}$ The diagonals ${AC}$ meets the segments ${DP}$ at ${Q.}$ The line through ${P}$ parallel to ${CD}$ mmets the extension of the side ${CB}$ beyond ${B}$ at ${K.}$ The line through ${Q}$ parallel to ${BD}$ meets the extension of the side ${CB}$ beyond ${B}$ at ${L.}$ Prove that the circumcircles of the triangles ${BKP}$ and ${CLQ}$ are tangent .

Problem 2. Determine whether there exist non-constant polynomials ${P(x)}$ and ${Q(x)}$ with real coefficients satisfying

$\displaystyle P(x)^{10}+P(x)^9 = Q(x)^{21}+Q(x)^{20}.$

Problem 3. Ann and Bob play a game on the edges of an infinite square grid, playing in turns. Ann plays the first move. A move consists of orienting any edge that has not yet been given an orientation. Bob wins if at any point a cycle has been created. Does Bob have a winning strategy?

Problem 4. Let ${a,b,c,d}$ be positive integers such that ${ad \neq bc}$ and ${gcd(a,b,c,d)=1}$ . Let ${S}$ be the set of values attained by ${\gcd(an+b,cn+d)}$ as ${n}$ runs through the positive integers. Show that ${S}$ is the set of all positive divisors of some positive integer.

Problem 5. Let ${n}$ be positive integer and fix ${2n}$ distinct points on a circle. Determine the number of ways to connect the points with ${n}$ arrows (oriented line segments) such that all of the following conditions hold:

each of the ${2n}$ points is a startpoint or endpoint of an arrow;
no two arrows intersect;
there are no two arrows ${\overrightarrow{AB}}$ and ${\overrightarrow{CD}}$ such that ${A}$ , ${B}$ , ${C}$ and ${D}$ appear in clockwise order around the circle (not necessarily consecutively).

Problem 6. Fix a circle ${\Gamma}$ , a line ${\ell}$ to tangent ${\Gamma}$ , and another circle ${\Omega}$ disjoint from ${\ell}$ such that ${\Gamma}$ and ${\Omega}$ lie on opposite sides of ${\ell}$ . The tangents to ${\Gamma}$ from a variable point ${X}$ on ${\Omega}$ meet ${\ell}$ at ${Y}$ and ${Z}$ . Prove that, as ${X}$ varies over ${\Omega}$ , the circumcircle of ${XYZ}$ is tangent to two fixed circles.

Source: Art of Problem Solving forums

Some quick ideas: For Problem 1 just consider the intersection of the circle ${(BKP)}$ with the circle ${(ABCD)}$ . You’ll notice immediately that this point belongs to the circle ${(CLQ)}$ . Furthermore, there is a common tangent to the two circles at this point.

For Problem 2 we have ${10\deg P = 21 \deg Q}$ . Eliminate the highest order term from both sides and look at the next one to get a contradiction.

Problem 4 becomes easy after noticing that if ${q}$ divides ${an+b}$ and ${cn+d}$ then ${q}$ divides ${ad-bc}$ .

In Problem 5 try to prove that the choice of start points determines that of the endpoints. Then you have a simple combinatorial proof.

Problem 6 is interesting and official solutions use inversions. Those are quite nice, but it may be worthwhile to understand what happens in the non-inverted configuration.

I will come back to some of these problems in some future posts.

Categories: Combinatorics, Geometry, IMO, Number theory, Olympiad Tags: Geometry, number theory, olympiad, Problem Solving, RMM

SEEMOUS 2018 – Problems

March 1, 2018 beni22sof Leave a comment

Problem 1. Let ${f:[0,1] \rightarrow (0,1)}$ be a Riemann integrable function. Show that

$\displaystyle \frac{\displaystyle 2\int_0^1 xf^2(x) dx }{\displaystyle \int_0^1 (f^2(x)+1)dx }< \frac{\displaystyle \int_0^1 f^2(x) dx}{\displaystyle \int_0^1 f(x)dx}.$

Problem 2. Let ${m,n,p,q \geq 1}$ and let the matrices ${A \in \mathcal M_{m,n}(\Bbb{R})}$ , ${B \in \mathcal M_{n,p}(\Bbb{R})}$ , ${C \in \mathcal M_{p,q}(\Bbb{R})}$ , ${D \in \mathcal M_{q,m}(\Bbb{R})}$ be such that

$\displaystyle A^t = BCD,\ B^t = CDA,\ C^t = DAB,\ D^t = ABC.$

Prove that ${(ABCD)^2 = ABCD}$ .

Problem 3. Let ${A,B \in \mathcal M_{2018}(\Bbb{R})}$ such that ${AB = BA}$ and ${A^{2018} = B^{2018} = I}$ , where ${I}$ is the identity matrix. Prove that if ${\text{tr}(AB) = 2018}$ then ${\text{tr}(A) = \text{tr}(B)}$ .

Problem 4. (a) Let ${f: \Bbb{R} \rightarrow \Bbb{R}}$ be a polynomial function. Prove that

$\displaystyle \int_0^\infty e^{-x} f(x) dx = f(0)+f'(0)+f''(0)+...$

(b) Let ${f}$ be a function which has a Taylor series expansion at ${0}$ with radius of convergence ${R=\infty}$ . Prove that if $\sum_{n=0}^\infty f^{(n)}(0)$ converges absolutely then $\int_0^{\infty} e^{-x} f(x)dx$ converges and

$\displaystyle \sum_{n=0}^\infty f^{(n)}(0) = \int_0^\infty e^{-x} f(x).$

Source: official site of SEEMOUS 2018

Hints: 1. Just use $2f(x) \leq f^2(x)+1$ and $xf^2(x) < f^2(x)$ . The strict inequality comes from the fact that the Riemann integral of strictly positive function cannot be equal to zero. This problem was too simple…

2. Use the fact that $ABCD = AA^t$ , therefore $ABCD$ is symmetric and positive definite. Next, notice that $(ABCD)^3 = ABCDABCDABCD = D^tC^tB^tA^t = (ABCD)^t = ABCD$ . Notice that $ABCD$ is diagonalizable and has eigenvalues among $-1,0,1$ . Since it is also positive definite, $-1$ cannot be an eigenvalue. This allows to conclude.

3. First note that the commutativity allows us to diagonalize $A,B$ using the same basis. Next, note that $A,B$ both have eigenvalues of modulus one. Then the trace of $AB$ is simply the sum $\sum \lambda_i\mu_i$ where $\lambda_i,\mu_i$ are eigenvalues of $A$ and $B$ , respectively. The fact that the trace equals $2018$ and the triangle inequality shows that eigenvalues of $A$ are a multiple of eigenvalues of $B$ . Finish by observing that they have the same eigenvalues.

4. (a) Integrate by parts and use a recurrence. (b) Use (a) and the approximation of a continuous function by polynomials on compacts to conclude.

I’m not sure about what others think, but the problems of this year seemed a bit too straightforward.

Categories: Algebra, Analysis, Higher Algebra, Inequalities Tags: 2018, olympiad, Problem Solving, SEEMOUS

Beni Bogoşel's blog

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IMO 2023 Problem 1

Problems of the International Mathematical Olympiad 2023

Applications of Helly’s theorem

IMO 2022 – Problem 4 – Geometry

Balkan Mathematical Olympiad 2022

Putnam 2021 – Day 1

Putnam 2020 problems – Day 1

IMC 2020 Day 1 – Some Hints for Problems 1-2

IMC 2020 Online – Problems – Day 1

IMC 2019 – Problems from Day 1

Putnam 2018 – Problem B2

IMO 2018 Problems – Day 1

Balkan Mathematical Olympiad 2018

Romanian Masters in Mathematics contest – 2018

SEEMOUS 2018 – Problems

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Beni Bogoşel

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