Found problems: 85335
2025 Bangladesh Mathematical Olympiad, P4
Let set $S$ be the smallest set of positive integers satisfying the following properties:
[list]
[*] $2$ is in set $S$.
[*] If $n^2$ is in set $S$, then $n$ is also in set $S$.
[*] If $n$ is in set $S$, then $(n+5)^2$ is also in set $S$.
[/list]
Determine which positive integers are not in set $S$.
2018 Hanoi Open Mathematics Competitions, 2
Let $f(x)$ be a polynomial such that $2f(x) + f(2 - x) = 5 + x$ for any real number x. Find the value of $f(0) + f(2)$.
A. $4$ B. $0$ C.$ 2$ D. $3$ E. $1$
2010 China Northern MO, 8
Let $x,y,z \in [0,1]$ , and $|y-z|\leq \frac{1}{2},|z-x|\leq \frac{1}{2},|x-y|\leq \frac{1}{2}$ . Find the maximum and minimum value of $W=x+y+z-yz-zx-xy$.
1992 Tournament Of Towns, (322) 3
A numismatist Fred has some coins. A diameter of any coin is no more than $10$ cm. All the coins are contained in a one-layer box of dimensions $30$ cm by $70$ cm. He is presented with a new coin. Its diameter is $25$ cm. Prove that it is possible to put all the coins in a one-layer box of dimensions $55$ cm by $55$ cm.
(Fedja Nazarov, St Petersburg)
Swiss NMO - geometry, 2005.1
Let $ABC$ be any triangle and $D, E, F$ the midpoints of $BC, CA, AB$. The medians $AD, BE$ and $CF$ intersect at point $S$. At least two of the quadrilaterals $AF SE, BDSF, CESD$ are cyclic. Show that the triangle $ABC$ is equilateral.
2025 Kyiv City MO Round 2, Problem 4
Let \( BE \) and \( CF \) be the medians of \( \triangle ABC \), and \( G \) be their intersection point. On segments \( GF \) and \( GE \), points \( K \) and \( L \), respectively, are chosen such that \( BK = CL = AG \). Prove that
\[
\angle BKF + \angle CLE = \angle BGC.
\]
[i]Proposed by Vadym Solomka[/i]
VI Soros Olympiad 1999 - 2000 (Russia), 10.4
Prove that the inequality $ r^2+r_a^2+r_b^2+ r_c^2 \ge 2S$ holds for an arbitrary triangle, where $r$ is the radius of the circle inscribed in the triangle, $r_a$, $r_b$, $r_c$ are the radii of its three excribed circles, $S$ is the area of the triangle.
2015 Czech-Polish-Slovak Junior Match, 2
We removed the middle square of $2 \times 2$ from the $8 \times 8$ board.
a) How many checkers can be placed on the remaining $60$ boxes so that there are no two not jeopardize?
b) How many at least checkers can be placed on the board so that they are at risk all $60$ squares?
(A lady is threatening the box she stands on, as well as any box she can get to in one move without going over any of the four removed boxes.)
2015 Hanoi Open Mathematics Competitions, 11
Given a convex quadrilateral $ABCD$. Let $O$ be the intersection point of diagonals $AC$ and $BD$ and let $I , K , H$ be feet of perpendiculars from $B , O , C$ to $AD$, respectively. Prove that $AD \times BI \times CH \le AC \times BD \times OK$.
2017 Iran MO (3rd round), 2
Let $ABCD$ be a trapezoid ($AB<CD,AB\parallel CD$) and $P\equiv AD\cap BC$. Suppose that $Q$ be a point inside $ABCD$ such that $\angle QAB=\angle QDC=90-\angle BQC$. Prove that $\angle PQA=2\angle QCD$.
2004 Belarusian National Olympiad, 5
Suppose that $A$ and $B$ are sets of real numbers such that
$$A\subset B+\alpha \mathbb{Z}\quad \text{and}\quad B\subset A+\alpha\mathbb{Z}\quad \text{for all}\quad \alpha>0$$
(where $X+\alpha\mathbb=\{x+\alpha n|x\in\mathbb{X}, n\in\mathbb{Z}\}$
(a) Does it follow that $A=B$
(b) The same question, with the assumption that $B$ is bounded
2009 Hanoi Open Mathematics Competitions, 1
Let $a,b, c$ be $3$ distinct numbers from $\{1, 2,3, 4, 5, 6\}$
Show that $7$ divides $abc + (7 - a)(7 - b)(7 - c)$
2010 Malaysia National Olympiad, 1
In the diagram, congruent rectangles $ABCD$ and $DEFG$ have a common vertex $D$. Sides $BC$ and $EF$ meet at $H$. Given that $DA = DE = 8$, $AB = EF = 12$, and $BH = 7$. Find the area of $ABHED$.
[img]https://cdn.artofproblemsolving.com/attachments/f/b/7225fa89097e7b20ea246b3aa920d2464080a5.png[/img]
VMEO II 2005, 6
For a given cyclic quadrilateral $ABCD$, let $I$ be a variable point on the diagonal $AC$ such that $I$ and $A$ are on the same side of the diagonal $BD$. Assume $E,F$ lie on the diagonal $BD$ such that $IE\parallel AB$ and $IF\parallel AD$. Show that $\angle BIE =\angle DCF $
2023 Romanian Master of Mathematics, 3
Let $n\geq 2$ be an integer and let $f$ be a $4n$-variable polynomial with real coefficients. Assume that, for any $2n$ points $(x_1,y_1),\dots,(x_{2n},y_{2n})$ in the Cartesian plane, $f(x_1,y_1,\dots,x_{2n},y_{2n})=0$ if and only if the points form the vertices of a regular $2n$-gon in some order, or are all equal.
Determine the smallest possible degree of $f$.
(Note, for example, that the degree of the polynomial $$g(x,y)=4x^3y^4+yx+x-2$$ is $7$ because $7=3+4$.)
[i]Ankan Bhattacharya[/i]
2007 Today's Calculation Of Integral, 223
Evaluate $ \int_{0}^{\pi}\sqrt{(\cos x\plus{}\cos 2x\plus{}\cos 3x)^{2}\plus{}(\sin x\plus{}\sin 2x\plus{}\sin 3x)^{2}}\ dx$.
1987 IMO Shortlist, 20
Let $n\ge2$ be an integer. Prove that if $k^2+k+n$ is prime for all integers $k$ such that $0\le k\le\sqrt{n\over3}$, then $k^2+k+n$ is prime for all integers $k$ such that $0\le k\le n-2$.[i](IMO Problem 6)[/i]
[b][i]Original Formulation[/i][/b]
Let $f(x) = x^2 + x + p$, $p \in \mathbb N.$ Prove that if the numbers $f(0), f(1), \cdots , f( \sqrt{p\over 3} )$ are primes, then all the numbers $f(0), f(1), \cdots , f(p - 2)$ are primes.
[i]Proposed by Soviet Union. [/i]
2023 Pan-American Girls’ Mathematical Olympiad, 4
In an acute-angled triangle $ABC$, let $D$ be a point on the segment $BC$. Let $R$ and $S$ be the feet of the perpendiculars from $D$ to $AC$ and $AB$, respectively. The line $DR$ intersects the circumcircle of $BDS$ at $X$, with $X \neq D$. Similarly, the line $DS$ intersects the circumcircle of $CDR$ at $Y$, with $Y \neq D$. Prove that if $XY$ is parallel to $RS$, then $D$ is the midpoint of $BC$.
2008 AMC 10, 22
Three red beads, two white beads, and one blue bead are placed in a line in random order. What is the probability that no two neighboring beads are the same color?
$ \textbf{(A)}\ \frac{1}{12} \qquad
\textbf{(B)}\ \frac{1}{10} \qquad
\textbf{(C)}\ \frac{1}{6} \qquad
\textbf{(D)}\ \frac{1}{3} \qquad
\textbf{(E)}\ \frac{1}{2}$
2000 Baltic Way, 6
Fredek runs a private hotel. He claims that whenever $ n \ge 3$ guests visit the hotel, it is possible to select two guests that have equally many acquaintances among the other guests, and that also have a common acquaintance or a common unknown among the guests. For which values of $ n$ is Fredek right? (Acquaintance is a symmetric relation.)
PEN G Problems, 10
Show that $\frac{1}{\pi} \arccos \left( \frac{1}{\sqrt{2003}} \right)$ is irrational.
2004 All-Russian Olympiad Regional Round, 10.1
The sum of positive numbers $a, b, c$ is equal to $\pi/2$. Prove that
$$\cos a + \cos b + \cos c > \sin a + \sin b + \sin c.$$
1989 AMC 12/AHSME, 27
Let $n$ be a positive integer. If the equation $2x+2y+z=n$ has $28$ solutions in positive integers $x, y$ and $z$, then $n$ must be either
$ \textbf{(A)}\ 14\ \text{or}\ 15 \qquad\textbf{(B)}\ 15\ \text{or}\ 16 \qquad\textbf{(C)}\ 16\ \text{or}\ 17 \qquad\textbf{(D)}\ 17\ \text{or}\ 18 \qquad\textbf{(E)}\ 18\ \text{or}\ 19 $
2024 CAPS Match, 2
For a positive integer $n$, an $n$-configuration is a family of sets $\left\langle A_{i,j}\right\rangle_{1\le i,j\le n}.$ An $n$-configuration is called [i]sweet[/i] if for every pair of indices $(i, j)$ with $1\le i\le n -1$ and $1\le j\le n$ we have $A_{i,j}\subseteq A_{i+1,j}$ and $A_{j,i}\subseteq A_{j,i+1}.$ Let $f(n, k)$ denote the number of sweet $n$-configurations such that $A_{n,n}\subseteq \{1, 2,\ldots , k\}$. Determine which number is larger: $f\left(2024, 2024^2\right)$ or $f\left(2024^2, 2024\right).$
1995 Rioplatense Mathematical Olympiad, Level 3, 6
A convex polygon with $2n$ sides is called [i]rhombic [/i] if its sides are equal and all pairs of opposite sides are parallel.
A rhombic polygon can be partitioned into rhombic quadrilaterals.
For what value of$ n$, a $2n$-sided rhombic polygon splits into $666$ rhombic quadrilaterals?