Found problems: 823
2021 IMO, 6
Let $m\ge 2$ be an integer, $A$ a finite set of integers (not necessarily positive) and $B_1,B_2,...,B_m$ subsets of $A$. Suppose that, for every $k=1,2,...,m$, the sum of the elements of $B_k$ is $m^k$. Prove that $A$ contains at least $\dfrac{m}{2}$ elements.
2002 Switzerland Team Selection Test, 7
Let $ABC$ be a triangle and $P$ an exterior point in the plane of the triangle. Suppose the lines $AP$, $BP$, $CP$ meet the sides $BC$, $CA$, $AB$ (or extensions thereof) in $D$, $E$, $F$, respectively. Suppose further that the areas of triangles $PBD$, $PCE$, $PAF$ are all equal. Prove that each of these areas is equal to the area of triangle $ABC$ itself.
2005 India IMO Training Camp, 3
Consider a matrix of size $n\times n$ whose entries are real numbers of absolute value not exceeding $1$. The sum of all entries of the matrix is $0$. Let $n$ be an even positive integer. Determine the least number $C$ such that every such matrix necessarily has a row or a column with the sum of its entries not exceeding $C$ in absolute value.
[i]Proposed by Marcin Kuczma, Poland[/i]
2018 District Olympiad, 2
Consider the set
\[M = \left\{
\begin{pmatrix}
a & b \\
c & d
\end{pmatrix}
\in\mathcal{M}_2(\mathbb{C})\ |\ ab = cd
\right\}.\]
a) Give an example of matrix $A\in M$ such that $A^{2017}\in M$ and $A^{2019}\in M$, but $A^{2018}\notin M$.
b) Show that if $A\in M$ and there exists the integer number $k\ge 1$ such that $A^k \in M$, $A^{k + 1}\in M$ si $A^{k + 2} \in M$, then $A^n\in M$, for any integer number $n\ge 1$.
2024 Mexican University Math Olympiad, 4
Given \( b > 0 \), consider the following matrix:
\[
B = \begin{pmatrix} b & b^2 \\ b^2 & b^3 \end{pmatrix}
\]
Denote by \( e_i \) the top left entry of \( B^i \). Prove that the following limit exists and calculate its value:
\[
\lim_{i \to \infty} \sqrt[i]{e_i}.
\]
2022 SEEMOUS, 3
Let $\alpha \in \mathbb{C}\setminus \{0\}$ and $A \in \mathcal{M}_n(\mathbb{C})$, $A \neq O_n$, be such that
$$A^2 + (A^*)^2 = \alpha A\cdot A^*,$$
where $A^* = (\bar A)^T.$ Prove that $\alpha \in \mathbb{R}$, $|\alpha| \le 2$. and $A\cdot A^* = A^*\cdot A.$
2019 LIMIT Category C, Problem 7
Let $O(4,\mathbb Z)$ be the set of all $4\times4$ orthogonal matrices over $\mathbb Z$, i.e., $A^tA=I=AA^t$. Then $|O(4,\mathbb Z)|$ is
2004 IMO Shortlist, 4
Consider a matrix of size $n\times n$ whose entries are real numbers of absolute value not exceeding $1$. The sum of all entries of the matrix is $0$. Let $n$ be an even positive integer. Determine the least number $C$ such that every such matrix necessarily has a row or a column with the sum of its entries not exceeding $C$ in absolute value.
[i]Proposed by Marcin Kuczma, Poland[/i]
1992 IMO Longlists, 34
Let $a, b, c$ be integers. Prove that there are integers $p_1, q_1, r_1, p_2, q_2, r_2$ such that
\[a = q_1r_2 - q_2r_1, b = r_1p_2 - r_2p_1, c = p_1q_2 - p_2q_1.\]
2010 Tournament Of Towns, 5
$33$ horsemen are riding in the same direction along a circular road. Their speeds are constant and pairwise distinct. There is a single point on the road where the horsemen can surpass one another. Can they ride in this fashion for arbitrarily long time ?
2023 Romania National Olympiad, 2
Let $A,B \in M_{n}(\mathbb{R}).$ Show that $rank(A) = rank(B)$ if and only if there exist nonsingular matrices $X,Y,Z \in M_{n}(\mathbb{R})$ such that
\[
AX + YB = AZB.
\]
2016 Romania National Olympiad, 1
Let be a $ 2\times 2 $ real matrix $ A $ that has the property that $ \left| A^d-I_2 \right| =\left| A^d+I_2 \right| , $ for all $ d\in\{ 2014,2016 \} . $
Prove that $ \left| A^n-I_2 \right| =\left| A^n+I_2 \right| , $ for any natural number $ n. $
2006 Miklós Schweitzer, 5
let $F_q$ be a finite field with char ≠ 2, and let $V = F_q \times F_q$ be the 2-dimensional vector space over $F_q$. Let L ⊂ V be a subset containing lines in all directions. The order of a point in V is the number of lines in L that pass through the point. Prove that L contains at least q lines having a third-order point.
2017 IMC, 8
Define the sequence $A_1,A_2,\ldots$ of matrices by the following recurrence: $$ A_1 = \begin{pmatrix} 0 & 1 \\ 1 & 0 \\ \end{pmatrix}, \quad A_{n+1} = \begin{pmatrix} A_n & I_{2^n} \\ I_{2^n} & A_n \\ \end{pmatrix} \quad (n=1,2,\ldots) $$ where $I_m$ is the $m\times m$ identity matrix.
Prove that $A_n$ has $n+1$ distinct integer eigenvalues $\lambda_0< \lambda_1<\ldots <\lambda_n$ with multiplicities $\binom{n}{0},\binom{n}{1},\ldots,\binom{n}{n}$, respectively.
2016 Vietnam Team Selection Test, 6
Given $16$ distinct real numbers $\alpha_1,\alpha_2,...,\alpha_{16}$. For each polynomial $P$, denote \[ V(P)=P(\alpha_1)+P(\alpha_2)+...+P(\alpha_{16}). \] Prove that there is a monic polynomial $Q$, $\deg Q=8$ satisfying:
i) $V(QP)=0$ for all polynomial $P$ has $\deg P<8$.
ii) $Q$ has $8$ real roots (including multiplicity).
2011 Morocco National Olympiad, 1
Solve the following equation in $\mathbb{R}^+$ :
\[\left\{\begin{matrix}
\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=2010\\
x+y+z=\frac{3}{670}
\end{matrix}\right.\]
2014 IMC, 1
Determine all pairs $(a, b)$ of real numbers for which there exists a unique symmetric $2\times 2$ matrix $M$ with real entries satisfying $\mathrm{trace}(M)=a$ and $\mathrm{det}(M)=b$.
(Proposed by Stephan Wagner, Stellenbosch University)
2008 Teodor Topan, 1
Solve in $ M_2(\mathbb{C})$ the equation $ X^2\equal{}\left(
\begin{array}{cc}
1 & 2 \\
3 & 6 \end{array}
\right)$
2011 Putnam, A6
Let $G$ be an abelian group with $n$ elements, and let \[\{g_1=e,g_2,\dots,g_k\}\subsetneq G\] be a (not necessarily minimal) set of distinct generators of $G.$ A special die, which randomly selects one of the elements $g_1,g_2,\dots,g_k$ with equal probability, is rolled $m$ times and the selected elements are multiplied to produce an element $g\in G.$
Prove that there exists a real number $b\in(0,1)$ such that \[\lim_{m\to\infty}\frac1{b^{2m}}\sum_{x\in G}\left(\mathrm{Prob}(g=x)-\frac1n\right)^2\] is positive and finite.
2018 Ramnicean Hope, 1
Let be a natural number $ n\ge 2, $ the real numbers $ a_1,a_2,\ldots ,a_n,b_1,b_2,\ldots, b_n, $ and the matrix defined as
$$ A=\left( a_i+b_j \right)_{1\le j\le n}^{1\le i\le n} . $$
[b]a)[/b] Show that $ n=2 $ if $ A $ is invertible.
[b]b)[/b] Prove that the pair of numbers $ a_1,a_2 $ and $ b_1,b_2 $ are both consecutive (not necessarily in this order), if $ A $ is an invertible matrix of integers whose inverse is a matrix of integers.
[i]Costică Ambrinoc[/i]
2021 Miklós Schweitzer, 1
Let $n, m \in \mathbb{N}$; $a_1,\ldots, a_m \in \mathbb{Z}^n$. Show that nonnegative integer linear combinations of these vectors give exactly the whole $\mathbb{Z}^n$ lattice, if $m \ge n$ and the following two statements are satisfied:
[list]
[*] The vectors do not fall into the half-space of $\mathbb{R}^n$ containing the origin (i.e. they do not fall on the same side of an $n-1$ dimensional subspace),
[*] the largest common divisor (not pairwise, but together) of $n \times n$ minor determinants of the matrix $(a_1,\ldots, a_m)$ (which is of size $m \times n$ and the $i$-th column is $a_i$ as a column vector) is $1$.
[/list]
2005 International Zhautykov Olympiad, 2
Let the circle $ (I; r)$ be inscribed in the triangle $ ABC$. Let $ D$ be the point of contact of this circle with $ BC$. Let $ E$ and $ F$ be the midpoints of $ BC$ and $ AD$, respectively. Prove that the three points $ I$, $ E$, $ F$ are collinear.
2003 Alexandru Myller, 3
Let be three elements $ a,b,c $ of a nontrivial, noncommutative ring, that satisfy $ ab=1-c, $ and such that there exists an element $ d $ from the ring such that $ a+cd $ is a unit. Prove that there exists an element $ e $ from the ring such that $ b+ec $ is a unit.
[i]Andrei Nedelcu[/i] and [i] Lucian Ladunca [/i]
1996 IMC, 9
Let $G$ be the subgroup of $GL_{2}(\mathbb{R})$ generated by $A$ and $B$, where
$$A=\begin{pmatrix}
2 &0\\
0&1
\end{pmatrix},\;
B=\begin{pmatrix}
1 &1\\
0&1
\end{pmatrix}$$.
Let $H$ consist of the matrices $\begin{pmatrix}
a_{11} &a_{12}\\
a_{21}& a_{22}
\end{pmatrix}$ in $G$ for which $a_{11}=a_{22}=1$.
a) Show that $H$ is an abelian subgroup of $G$.
b) Show that $H$ is not finitely generated.
2007 IberoAmerican Olympiad For University Students, 4
Consider an infinite sequence $a_1,a_2,\cdots$ whose terms all belong to $\left\{1,2\right\}$. A positive integer with $n$ digits is said to be [i]good[/i] if its decimal representation has the form $a_ra_{r+1}\cdots a_{r+(n-1)}$, for some positive integer $r$. Suppose that there are at least $2008$ [i]good[/i] numbers with a million digits. Prove that there are at least $2008$ [i]good[/i] numbers with $2007$ digits.