Found problems: 148
2021 Junior Balkan Team Selection Tests - Romania, P3
Let $p,q$ be positive integers. For any $a,b\in\mathbb{R}$ define the sets $$P(a)=\bigg\{a_n=a \ + \ n \ \cdot \ \frac{1}{p} : n\in\mathbb{N}\bigg\}\text{ and }Q(b)=\bigg\{b_n=b \ + \ n \ \cdot \ \frac{1}{q} : n\in\mathbb{N}\bigg\}.$$
The [i]distance[/i] between $P(a)$ and $Q(b)$ is the minimum value of $|x-y|$ as $x\in P(a), y\in Q(b)$. Find the maximum value of the distance between $P(a)$ and $Q(b)$ as $a,b\in\mathbb{R}$.
2006 Romania Team Selection Test, 2
Let $ABC$ be a triangle with $\angle B = 30^{\circ }$. We consider the closed disks of radius $\frac{AC}3$, centered in $A$, $B$, $C$. Does there exist an equilateral triangle with one vertex in each of the 3 disks?
[i]Radu Gologan, Dan Schwarz[/i]
2022 District Olympiad, P1
Let $e$ be the identity of monoid $(M,\cdot)$ and $a\in M$ an invertible element. Prove that
[list=a]
[*]The set $M_a:=\{x\in M:ax^2a=e\}$ is nonempty;
[*]If $b\in M_a$ is invertible, then $b^{-1}\in M_a$ if and only if $a^4=e$;
[*]If $(M_a,\cdot)$ is a monoid, then $x^2=e$ for all $x\in M_a.$
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[i]Mathematical Gazette[/i]
2022 Romania Team Selection Test, 4
Any positive integer $N$ which can be expressed as the sum of three squares can obviously be written as \[N=\frac{a^2+b^2+c^2+d^2}{1+abcd}\]where $a,b,c,d$ are nonnegative integers. Is the mutual assertion true?
2022 Romania Team Selection Test, 1
Let $ABC$ be an acute scalene triangle and let $\omega$ be its Euler circle. The tangent $t_A$ of $\omega$ at the foot of the height $A$ of the triangle ABC, intersects the circle of diameter $AB$ at the point $K_A$ for the second time. The line determined by the feet of the heights $A$ and $C$ of the triangle $ABC$ intersects the lines $AK_A$ and $BK_A$ at the points $L_A$ and $M_A$, respectively, and the lines $t_A$ and $CM_A$ intersect at the point $N_A$.
Points $K_B, L_B, M_B, N_B$ and $K_C, L_C, M_C, N_C$ are defined similarly for $(B, C, A)$ and $(C, A, B)$ respectively. Show that the lines $L_AN_A, L_BN_B,$ and $L_CN_C$ are concurrent.
2022 Romania National Olympiad, P3
Let $f,g:\mathbb{R}\to\mathbb{R}$ be two nondecreasing functions.
[list=a]
[*]Show that for any $a\in\mathbb{R},$ $b\in[f(a-0),f(a+0)]$ and $x\in\mathbb{R},$ the following inequality holds \[\int_a^xf(t) \ dt\geq b(x-a).\]
[*]Given that $[f(a-0),f(a+0)]\cap[g(a-0),g(a+0)]\neq\emptyset$ for any $a\in\mathbb{R},$ prove that for any real numbers $a<b$\[\int_a^b f(t) \ dt=\int_a^b g(t) \ dt.\]
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[i]Note: $h(a-0)$ and $h(a+0)$ denote the limits to the left and to the right respectively of a function $h$ at point $a\in\mathbb{R}.$[/i]
2022 Romania EGMO TST, P2
On a board there is a regular polygon $A_1A_2\ldots A_{99}.$ Ana and Barbu alternatively occupy empty vertices of the polygon and write down triangles on a list: Ana only writes obtuse triangles, while Barbu only writes acute ones.
At the first turn, Ana chooses three vertices $X,Y$ and $Z$ and writes down $\triangle XYZ.$ Then, Barbu chooses two of $X,Y$ and $Z,$ for example $X$ and $Y$, and an unchosen vertex $T$, and writes down $\triangle XYT.$ The game goes on and at each turn, the player must choose a new vertex $R$ and write down $\triangle PQR$, where $P$ is the last vertex chosen by the other player, and $Q$ is one of the other vertices of the last triangle written down by the other player.
If one player cannot perform a move, then the other one wins. If both people play optimally, determine who has a winning strategy.
2022 District Olympiad, P4
Let $A\in\mathcal{M}_n(\mathbb{C})$ where $n\geq 2.$ Prove that if $m=|\{\text{rank}(A^k)-\text{rank}(A^{k+1})":k\in\mathbb{N}^*\}|$ then $n+1\geq m(m+1)/2.$
2020 Romania EGMO TST, P1
Determine if for any positive integers $a,b,c$ there exist pairwise distinct non-negative integers $A,B,C$ which are greater than $2019$ such that $a+A,b+B$ and $c+C$ divide $ABC$.
2017 Danube Mathematical Olympiad, 2
Let $n\geq 3$ be a positive integer. Consider an $n\times n$ square. In each cell of the square, one of the numbers from the set $M=\{1,2,\ldots,2n-1\}$ is to be written. One such filling is called [i]good[/i] if, for every index $1\leq i\leq n,$ row no. $i$ and column no. $i,$ together, contain all the elements of $M$.
[list=a]
[*]Prove that there exists $n\geq 3$ for which a good filling exists.
[*]Prove that for $n=2017$ there is no good filling of the $n\times n$ square.
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2020 Stars of Mathematics, 2
Let $ABC$ be a triangle, let $I$ be its incentre and let $D$ be the orthogonal projection of $I$ on $BC.$ The circle $\odot(ABC)$ crosses the line $AI$ again at $M,$ and the line $DM$ again at $N.$ Prove that the lines $AN$ and $IN$ are perpendicular.
[i]Freddie Illingworth & Dominic Yeo[/i]
2021 Junior Balkan Team Selection Tests - Romania, P2
Let $O$ be the circumcenter of triangle $ABC$ and let $AD$ be the height from $A$ ($D\in BC$). Let $M,N,P$ and $Q$ be the midpoints of $AB,AC,BD$ and $CD$ respectively. Let $\mathcal{C}_1$ and $\mathcal{C}_2$ be the circumcircles of triangles $AMN$ and $POQ$. Prove that $\mathcal{C}_1\cap \mathcal{C}_2\cap AD\neq \emptyset$.
2017 Romania Team Selection Test, P1
Let $ABCD$ be a trapezium, $AD\parallel BC$, and let $E,F$ be points on the sides$AB$ and $CD$, respectively. The circumcircle of $AEF$ meets $AD$ again at $A_1$, and the circumcircle of $CEF$ meets $BC$ again at $C_1$. Prove that $A_1C_1,BD,EF$ are concurrent.
2021 Romania Team Selection Test, 3
Let $\mathcal{P}$ be a convex quadrilateral. Consider a point $X$ inside $\mathcal{P}.$ Let $M,N,P,Q$ be the projections of $X$ on the sides of $\mathcal{P}.$ We know that $M,N,P,Q$ all sit on a circle of center $L.$ Let $J$ and $K$ be the midpoints of the diagonals of $\mathcal{P}.$ Prove that $J,K$ and $L$ lie on a line.
1978 Romania Team Selection Test, 6
Show that there is no polyhedron whose projection on the plane is a nondegenerate triangle.
2021 Winter Stars of Mathematics, 3
Determine all integers $n>1$ whose positive divisors add up to a power of $3.$
[i]Andrei Bâra[/i]
2022 District Olympiad, P3
Let $(x_n)_{n\geq 1}$ be the sequence defined recursively as such: \[x_1=1, \ x_{n+1}=\frac{x_1}{n+1}+\frac{x_2}{n+2}+\cdots+\frac{x_n}{2n} \ \forall n\geq 1.\]Consider the sequence $(y_n)_{n\geq 1}$ such that $y_n=(x_1^2+x_2^2+\cdots x_n^2)/n$ for all $n\geq 1.$ Prove that
[list=a]
[*]$x_{n+1}^2<y_n/2$ and $y_{n+1}<(2n+1)/(2n+2)\cdot y_n$ for all $n\geq 1;$
[*]$\lim_{n\to\infty}x_n=0.$
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2022 Romania National Olympiad, P3
Determine all functions $f:\mathbb{R}\to\mathbb{R}$ which are differentiable in $0$ and satisfy the following inequality for all real numbers $x,y$ \[f(x+y)+f(xy)\geq f(x)+f(y).\][i]Dan Ștefan Marinescu and Mihai Piticari[/i]
2022 Romania National Olympiad, P1
Let $a\neq 1$ be a positive real number. Find all real solutions to the equation $a^x=x^x+\log_a(\log_a(x)).$
[i]Mihai Opincariu[/i]
2018 IMAR Test, 3
Let $S$ be a finite set and let $\mathcal{P}(S)$ be its power set, i.e., the set of all subsets of $S$, the empty set and $S$, inclusive. If $\mathcal{A}$ and $\mathcal{B}$ are non-empty subsets of $\mathcal{P}(S),$ let \[\mathcal{A}\vee \mathcal{B}=\{X:X\subseteq A\cup B,A\in\mathcal{A},B\in\mathcal{B}\}.\] Given a non-negative integer $n\leqslant |S|,$ determine the minimal size $\mathcal{A}\vee \mathcal{B}$ may have, where $\mathcal{A}$ and $\mathcal{B}$ are non-empty subsets of $\mathcal{P}(S)$ such that $|\mathcal{A}|+|\mathcal{B}|>2^n$.
[i]Amer. Math. Monthly[/i]
2022 District Olympiad, P3
Find all values of $n\in\mathbb{N}^*$ for which \[I_n:=\int_0^\pi\cos(x)\cdot\cos(2x)\cdot\ldots\cdot\cos(nx) \ dx=0.\]
2022 Romania Team Selection Test, 3
Let $n\geq 2$ be an integer. Let $a_{ij}, \ i,j=1,2,\ldots,n$ be $n^2$ positive real numbers satisfying the following conditions:
[list=1]
[*]For all $i=1,\ldots,n$ we have $a_{ii}=1$ and,
[*]For all $j=2,\ldots,n$ the numbers $a_{ij}, \ i=1,\ldots, j-1$ form a permutation of $1/a_{ji}, \ i=1,\ldots, j-1.$
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Given that $S_i=a_{i1}+\cdots+a_{in}$, determine the maximum value of the sum $1/S_1+\cdots+1/S_n.$
2022 Romania Team Selection Test, 2
Fix a nonnegative integer $a_0$ to define a sequence of integers $a_0,a_1,\ldots$ by letting $a_k,k\geq 1$ be the smallest integer (strictly) greater than $a_{k-1}$ making $a_{k-1}+a_k{}$ into a perfect square. Let $S{}$ be the set of positive integers not expressible as the difference of two terms of the sequence $(a_k)_{k\geq 0}.$ Prove that $S$ is finite and determine its size in terms of $a_0.$
2000 District Olympiad (Hunedoara), 1
Solve in the set of $ 2\times 2 $ integer matrices the equation
$$ X^2-4\cdot X+4\cdot\left(\begin{matrix}1\quad 0\\0\quad 1\end{matrix}\right) =\left(\begin{matrix}7\quad 8\\12\quad 31\end{matrix}\right) . $$
2022 Junior Balkan Team Selection Tests - Romania, P4
For any $n$-tuple $a=(a_1,a_2,\ldots,a_n)\in\mathbb{N}_0^n$ of nonnegative integers, let $d_a$ denote the number of pairs of indices $(i,j)$ such that $a_i-a_j=1.$ Determine the maximum possible value of $d_a$ as $a$ ranges over all elements of $\mathbb{N}_0^n.$