Found problems: 85335
2023 BMT, 10
Let $a$ denote the positive real root of the polynomial $x^2 -3x-2$. Compute the remainder when $\lfloor a^{1000}\rfloor $ is divided by the prime number $997$. Here, $\lfloor r\rfloor$ denotes the greatest integer less than $r$.
2005 QEDMO 1st, 8 (Z2)
Prove that if $n$ can be written as $n=a^2+ab+b^2$, then also $7n$ can be written that way.
2020 IOM, 3
Let $n>1$ be a given integer. The Mint issues coins of $n$ different values $a_1, a_2, ..., a_n$, where each $a_i$ is a positive integer (the number of coins of each value is unlimited). A set of values $\{a_1, a_2,..., a_n\}$ is called [i]lucky[/i], if the sum $a_1+ a_2+...+ a_n$ can be collected in a unique way (namely, by taking one coin of each value).
(a) Prove that there exists a lucky set of values $\{a_1, a_2, ..., a_n\}$ with $$a_1+ a_2+...+ a_n < n \cdot 2^n.$$
(b) Prove that every lucky set of values $\{a_1, a_2,..., a_n\}$ satisfies $$a_1+ a_2+...+ a_n >n \cdot 2^{n-1}.$$
Proposed by Ilya Bogdanov
1954 Czech and Slovak Olympiad III A, 3
Show that $$\log_2\pi+\log_4\pi<\frac52.$$
2018 Singapore Senior Math Olympiad, 3
Determine the largest positive integer $n$ such that the following statement is true:
There exists $n$ real polynomials, $P_1(x),\ldots,P_n(x)$ such that the sum of any two of them have no real roots but the sum of any three does.
2007 Princeton University Math Competition, 2
In how many distinguishable ways can $10$ distinct pool balls be formed into a pyramid ($6$ on the bottom, $3$ in the middle, one on top), assuming that all rotations of the pyramid are indistinguishable?
2007 Iran Team Selection Test, 2
Let $A$ be the largest subset of $\{1,\dots,n\}$ such that for each $x\in A$, $x$ divides at most one other element in $A$. Prove that \[\frac{2n}3\leq |A|\leq \left\lceil \frac{3n}4\right\rceil. \]
2020 DMO Stage 1, 3.
[b]Q.[/b] Determine all the functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that
$$f(x) \geqslant x+1, \forall\ x \in \mathbb{R}\quad \text{and}\quad f(x+y) \geqslant f(x) f(y), \forall\ x, y \in \mathbb{R}$$
[i]Proposed by TuZo[/i]
2009 Junior Balkan Team Selection Test, 4
In the decimal expression of a $ 2009$-digit natural number there are only the digits $ 5$ and $ 8$. Prove that we can get a $ 2008$-digit number divisible by $ 11$ if we remove just one digit from the number.
1978 IMO Longlists, 4
Two identically oriented equilateral triangles, $ABC$ with center $S$ and $A'B'C$, are given in the plane. We also have $A' \neq S$ and $B' \neq S$. If $M$ is the midpoint of $A'B$ and $N$ the midpoint of $AB'$, prove that the triangles $SB'M$ and $SA'N$ are similar.
1937 Moscow Mathematical Olympiad, 036
* Given a regular dodecahedron. Find how many ways are there to draw a plane through it so that its section of the dodecahedron is a regular hexagon?
2013 Iran Team Selection Test, 11
Let $a,b,c$ be sides of a triangle such that $a\geq b \geq c$. prove that:
$\sqrt{a(a+b-\sqrt{ab})}+\sqrt{b(a+c-\sqrt{ac})}+\sqrt{c(b+c-\sqrt{bc})}\geq a+b+c$
2013 Bangladesh Mathematical Olympiad, 6
There are $n$ cities in a country. Between any two cities there is at most one road. Suppose that the total number of roads is $n.$ Prove that there is a city such that starting from there it is possible to come back to it without ever travelling the same road twice.
Geometry Mathley 2011-12, 12.4
Quadrilateral$ ABCD$ has two diagonals $AC,BD$ that are mutually perpendicular. Let $M$ be the Miquel point of the complete quadrilateral formed by lines $AB,BC,CD,DA$. Suppose that $L$ is the intersection of two circles $(MAC)$ and $(MBD)$. Prove that the circumcenters of triangles $LAB,LBC,LCD,LDA$ are on the same circle called $\omega$ and that three circles $(MAC), (MBD), \omega$ are pairwise orthogonal.
Nguyễn Văn Linh
1987 Flanders Math Olympiad, 3
Find all continuous functions $f: \mathbb{R}\rightarrow\mathbb{R}$ such that \[f(x)^3 = -\frac x{12}\cdot\left(x^2+7x\cdot f(x)+16\cdot f(x)^2\right),\ \forall x \in \mathbb{R}.\]
2023 Durer Math Competition Finals, 7
The area of a rectangle is $64$ cm$^2$, and the radius of its circumscribed circle is $7$ cm. What is the perimeter of the rectangle in centimetres?
1953 Kurschak Competition, 3
$ABCDEF$ is a convex hexagon with all its sides equal. Also $\angle A + \angle C + \angle E = \angle B + \angle D + \angle F$. Show that $\angle A = \angle D$, $\angle B = \angle E$ and $\angle C = \angle F$.
2016 IMO Shortlist, N3
A set of positive integers is called [i]fragrant[/i] if it contains at least two elements and each of its elements has a prime factor in common with at least one of the other elements. Let $P(n)=n^2+n+1$. What is the least possible positive integer value of $b$ such that there exists a non-negative integer $a$ for which the set $$\{P(a+1),P(a+2),\ldots,P(a+b)\}$$ is fragrant?
2014 NIMO Problems, 6
We know $\mathbb Z_{210} \cong \mathbb Z_2 \times \mathbb Z_3 \times \mathbb Z_5 \times \mathbb Z_7$.
Moreover,\begin{align*}
53 & \equiv 1 \pmod{2} \\
53 & \equiv 2 \pmod{3} \\
53 & \equiv 3 \pmod{5} \\
53 & \equiv 4 \pmod{7}.
\end{align*}
Let
\[ M = \left(
\begin{array}{ccc}
53 & 158 & 53 \\
23 & 93 & 53 \\
50 & 170 & 53
\end{array}
\right). \]
Based on the above, find $\overline{(M \mod{2})(M \mod{3})(M \mod{5})(M \mod{7})}$.
2001 JBMO ShortLists, 13
At a conference there are $n$ mathematicians. Each of them knows exactly $k$ fellow mathematicians. Find the smallest value of $k$ such that there are at least three mathematicians that are acquainted each with the other two.
[color=#BF0000]Rewording of the last line for clarification:[/color]
Find the smallest value of $k$ such that there (always) exists $3$ mathematicians $X,Y,Z$ such that $X$ and $Y$ know each other, $X$ and $Z$ know each other and $Y$ and $Z$ know each other.
2019 ELMO Shortlist, C1
Elmo and Elmo's clone are playing a game. Initially, $n\geq 3$ points are given on a circle. On a player's turn, that player must draw a triangle using three unused points as vertices, without creating any crossing edges. The first player who cannot move loses. If Elmo's clone goes first and players alternate turns, who wins? (Your answer may be in terms of $n$.)
[i]Proposed by Milan Haiman[/i]
2022 Kosovo National Mathematical Olympiad, 4
Find all positive integers $k,m$ and $n$ such that $k!+3^m=3^n$
2024 USA TSTST, 6
Determine whether there exists a function $f: \mathbb{Z}_{> 0} \rightarrow \mathbb{Z}_{> 0}$ such that for all positive integers $m$ and $n$,
\[f(m+nf(m))=f(n)^m+2024! \cdot m.\]
[i]Jaedon Whyte[/i]
2011 Today's Calculation Of Integral, 767
For $0\leq t\leq 1$, define $f(t)=\int_0^{2\pi} |\sin x-t|dx.$
Evaluate $\int_0^1 f(t)dt.$
2005 Irish Math Olympiad, 5
Let $ a,b,c$ be nonnegative real numbers. Prove that:
$ \frac{1}{3}((a\minus{}b)^2\plus{}(b\minus{}c)^2\plus{}(c\minus{}a)^2) \le a^2\plus{}b^2\plus{}c^2\minus{}3 \sqrt[3]{a^2 b^2 c^2 } \le (a\minus{}b)^2\plus{}(b\minus{}c)^2\plus{}(c\minus{}a)^2.$