Found problems: 85335
2023 Austrian MO National Competition, 1
Let $a, b, c, d$ be positive reals strictly smaller than $1$, such that $a+b+c+d=2$. Prove that $$\sqrt{(1-a)(1-b)(1-c)(1-d)} \leq \frac{ac+bd}{2}. $$
2022 Stanford Mathematics Tournament, 4
Let the roots of
\[x^{2022}-7x^{2021}+8x^2+4x+2\]
be $r_1,r_2,\dots,r_{2022}$, the roots of
\[x^{2022}-8x^{2021}+27x^2+9x+3\]
be $s_1,s_2,\dots,s_{2022}$, and the roots of
\[x^{2022}-9x^{2021}+64x^2+16x+4\]
be $t_1,t_2,\dots,t_{2022}$. Compute the value of
\[\sum_{1\le i,j\le2022}r_is_j+\sum_{1\le i,j\le2022}s_it_j+\sum_{1\le i,j\le2022}t_ir_j.\]
2016 Junior Balkan Team Selection Tests - Moldova, 3
Let $ABC$ be an isosceles triangle with $\measuredangle C=\measuredangle B=36$. The point $M$ is in interior of $ ABC$ such that $\measuredangle MBC=24^{\circ} , \measuredangle BCM=30^{\circ}$ $N = AM \cap BC.$. Find $\measuredangle MCB$ .
2020 Italy National Olympiad, #4
Let $ABC$ be an acute-angled triangle with $AB=AC$, let $D$ be the foot of perpendicular, of the point $C$, to the line $AB$ and the point $M$ is the midpoint of $AC$. Finally, the point $E$ is the second intersection of the line $BC$ and the circumcircle of $\triangle CDM$. Prove that the lines $AE, BM$ and $CD$ are concurrents if and only if $CE=CM$.
Kvant 2020, M2594
It is known that for some $x{}$ and $y{}$ the sums $\sin x+ \cos y$ and $\sin y + \cos x$ are positive rational numbers. Prove that there exist natural numbers $m{}$ and $n{}$ such that $m\sin x+n\cos x$ is a natural number.
[i]Proposed by N. Agakhanov[/i]
1976 Bundeswettbewerb Mathematik, 3
A set $S$ of rational numbers is ordered in a tree-diagram in such a way that each rational number $\frac{a}{b}$ (where $a$ and $b$ are coprime integers) has exactly two successors: $\frac{a}{a+b}$ and $\frac{b}{a+b}$. How should the initial element be selected such that this tree contains the set of all rationals $r$ with $0 < r < 1$? Give a procedure for determining the level of a rational number $\frac{p}{q}$ in this tree.
2010 Iran MO (3rd Round), 5
[b]interesting sequence[/b]
$n$ is a natural number and $x_1,x_2,...$ is a sequence of numbers $1$ and $-1$ with these properties:
it is periodic and its least period number is $2^n-1$. (it means that for every natural number $j$ we have $x_{j+2^n-1}=x_j$ and $2^n-1$ is the least number with this property.)
There exist distinct integers $0\le t_1<t_2<...<t_k<n$ such that for every natural number $j$ we have
\[x_{j+n}=x_{j+t_1}\times x_{j+t_2}\times ... \times x_{j+t_k}\]
Prove that for every natural number $s$ that $s<2^n-1$ we have
\[\sum_{i=1}^{2^n-1}x_ix_{i+s}=-1\]
Time allowed for this question was 1 hours and 15 minutes.
2014 PUMaC Combinatorics B, 2
A $100 \times 100$ grid is given as shown. We choose a certain number of cells such that exactly two cells in each row and column are selected. Find the sum of numbers in these cells.
1989 Chile National Olympiad, 4
The vault of a bank has $N$ locks. To open it, they must be operated simultaneously. Five executives have some of the keys, so any trio can open the vault, but no pair can do it. Determine $N$.
2019 Saint Petersburg Mathematical Olympiad, 2
On the blackboard there are written $100$ different positive integers . To each of these numbers is added the $\gcd$ of the $99$ other numbers . In the new $100$ numbers , is it possible for $3$ of them to be equal.
[i] (С. Берлов)[/i]
1989 IMO Shortlist, 8
Let $ R$ be a rectangle that is the union of a finite number of rectangles $ R_i,$ $ 1 \leq i \leq n,$ satisfying the following conditions:
[b](i)[/b] The sides of every rectangle $ R_i$ are parallel to the sides of $ R.$
[b](ii)[/b] The interiors of any two different rectangles $ R_i$ are disjoint.
[b](iii)[/b] Each rectangle $ R_i$ has at least one side of integral length.
Prove that $ R$ has at least one side of integral length.
[i]Variant:[/i] Same problem but with rectangular parallelepipeds having at least one integral side.
2001 Stanford Mathematics Tournament, 15
Let $ABC$ be an isosceles triangle with $\angle{ABC} = \angle{ACB} = 80^\circ$. Let $D$ be a point on $AB$ such that $\angle{DCB} = 60^\circ$ and $E$ be a point on $AC$ such that $\angle{ABE} = 30^\circ$. Find $\angle{CDE}$ in degrees.
Kyiv City MO Seniors Round2 2010+ geometry, 2011.11.4
Let three circles be externally tangent in pairs, with parallel diameters $A_1A_2, B_1B_2, C_1C_2$ (i.e. each of the quadrilaterals $A_1B_1B_2A_2$ and $A_1C_1C_2A_2$ is a parallelogram or trapezoid, which segment $A_1A_2$ is the base). Prove that $A_1B_2, B_1C_2, C_1A_2$ intersect at one point.
(Yuri Biletsky )
2007 China Team Selection Test, 1
Points $ A$ and $ B$ lie on the circle with center $ O.$ Let point $ C$ lies outside the circle; let $ CS$ and $ CT$ be tangents to the circle. $ M$ be the midpoint of minor arc $ AB$ of $ (O).$ $ MS,\,MT$ intersect $ AB$ at points $ E,\,F$ respectively. The lines passing through $ E,\,F$ perpendicular to $ AB$ cut $ OS,\,OT$ at $ X$ and $ Y$ respectively.
A line passed through $ C$ intersect the circle $ (O)$ at $ P,\,Q$ ($ P$ lies on segment $ CQ$). Let $ R$ be the intersection of $ MP$ and $ AB,$ and let $ Z$ be the circumcentre of triangle $ PQR.$
Prove that: $ X,\,Y,\,Z$ are collinear.
2010 Korea Junior Math Olympiad, 6
Let $n\in\mathbb{N}$ and $p$ is the odd prime number. Define the sequence $a_n$ such that $a_1=pn+1$ and $a_{k+1}=na_k+1$ for all $k \in \mathbb{N}$ . Prove that $a_{p-1}$ is compound number.
2024 Azerbaijan National Mathematical Olympiad, 5
In a scalene triangle $ABC$, the points $E$ and $F$ are the foot of altitudes drawn from $B$ and $C$, respectively. The points $X$ and $Y$ are the reflections of the vertices $B$ and $C$ to the line $EF$, respectively. Let the circumcircles of the $\triangle ABC$ and $\triangle AEF$ intersect at $T$ for the second time. Show that the four points $A, X, Y, T$ lie on a single circle.
MIPT student olimpiad autumn 2022, 3
How many ways are there (in terms of power) to represent the number 1 as a finite number
or an infinite sum of some subset of the set:
{$\phi^{-n} | n \in Z^+$}
$\phi=\frac{1+\sqrt5}{2}$
2004 Greece Junior Math Olympiad, 1
The numbers $203$ and $298$ divided with the positive integer $x$ give both remainder $13$. Which are the possible values of $x$ ?
1960 AMC 12/AHSME, 30
Given the line $3x+5y=15$ and a point on this line equidistant from the coordinate axes. Such a point exists in:
$ \textbf{(A)}\ \text{none of the quadrants} \qquad\textbf{(B)}\ \text{quadrant I only} \qquad\textbf{(C)}\ \text{quadrants I, II only} \qquad$
$\textbf{(D)}\ \text{quadrants I, II, III only} \qquad\textbf{(E)}\ \text{each of the quadrants} $
2012 Indonesia TST, 4
Find all odd prime $p$ such that $1+k(p-1)$ is prime for all integer $k$ where $1 \le k \le \dfrac{p-1}{2}$.
1994 Argentina National Olympiad, 6
A $9\times 9$ board has a number written on each square: all squares in the first row have $1$, all squares in the second row have $2$, $\ldots$, all squares in the ninth row have $9$.
We will call [i]special [/i] rectangle any rectangle of $2\times 3$ or $3\times 2$ or $4\times 5$ or $5\times 4$ on the board.
The permitted operations are:
$\bullet$ Simultaneously add $1$ to all the numbers located in a special rectangle.
$\bullet$ Simultaneously subtract $1$ from all numbers located in a special rectangle.
Demonstrate that it is possible to achieve, through a succession of permitted operations, that $80$ squares to have $0$ (zero). What number is left in the remaining box?
2014 Israel National Olympiad, 5
Let $p$ be a polynomial with integer coefficients satisfying $p(16)=36,p(14)=16,p(5)=25$. Determine all possible values of $p(10)$.
2023 Thailand October Camp, 5
Let $ABC$ be an acute-angled triangle with $AC > AB$, let $O$ be its circumcentre, and let $D$ be a point on the segment $BC$. The line through $D$ perpendicular to $BC$ intersects the lines $AO, AC,$ and $AB$ at $W, X,$ and $Y,$ respectively. The circumcircles of triangles $AXY$ and $ABC$ intersect again at $Z \ne A$.
Prove that if $W \ne D$ and $OW = OD,$ then $DZ$ is tangent to the circle $AXY.$
1969 Putnam, B4
Show that any curve of unit length can be covered by a closed rectangle of area $1 \slash 4$.
2014 AMC 10, 7
Nonzero real numbers $x$, $y$, $a$, and $b$ satisfy $x < a$ and $y < b$. How many of the following inequalities must be true?
(I) $x+y < a+b$
(II) $x-y < a-b$
(III) $xy < ab$
(IV) $\frac{x}{y} < \frac{a}{b}$
${ \textbf{(A)}\ 0\qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}}\ 3\qquad\textbf{(E)}\ 4$