Found problems: 85335
2017 Miklós Schweitzer, 8
Let the base $2$ representation of $x\in[0;1)$ be $x=\sum_{i=0}^\infty \frac{x_i}{2^{i+1}}$. (If $x$ is dyadically rational, i.e. $x\in\left\{\frac{k}{2^n}\,:\, k,n\in\mathbb{Z}\right\}$, then we choose the finite representation.) Define function $f_n:[0;1)\to\mathbb{Z}$ by
$$f_n(x)=\sum_{j=0}^{n-1}(-1)^{\sum_{i=0}^j x_i}.$$Does there exist a function $\varphi:[0;\infty)\to[0;\infty)$ such that $\lim_{x\to\infty} \varphi(x)=\infty$ and
$$\sup_{n\in\mathbb{N}}\int_0^1 \varphi(|f_n(x)|)\mathrm{d}x<\infty\, ?$$
1958 AMC 12/AHSME, 7
A straight line joins the points $ (\minus{}1,1)$ and $ (3,9)$. Its $ x$-intercept is:
$ \textbf{(A)}\ \minus{}\frac{3}{2}\qquad
\textbf{(B)}\ \minus{}\frac{2}{3}\qquad
\textbf{(C)}\ \frac{2}{5}\qquad
\textbf{(D)}\ 2\qquad
\textbf{(E)}\ 3$
1980 Yugoslav Team Selection Test, Problem 2
Let $a,b,c,m$ be integers, where $m>1$. Prove that if
$$a^n+bn+c\equiv0\pmod m$$for each natural number $n$, then $b^2\equiv0\pmod m$. Must $b\equiv0\pmod m$ also hold?
2004 Germany Team Selection Test, 2
Three distinct points $A$, $B$, and $C$ are fixed on a line in this order. Let $\Gamma$ be a circle passing through $A$ and $C$ whose center does not lie on the line $AC$. Denote by $P$ the intersection of the tangents to $\Gamma$ at $A$ and $C$. Suppose $\Gamma$ meets the segment $PB$ at $Q$. Prove that the intersection of the bisector of $\angle AQC$ and the line $AC$ does not depend on the choice of $\Gamma$.
2007 Federal Competition For Advanced Students, Part 2, 2
Find all tuples $ (x_1,x_2,x_3,x_4,x_5,x_6)$ of non-negative integers, such that the following system of equations holds:
$ x_1x_2(1\minus{}x_3)\equal{}x_4x_5 \\
x_2x_3(1\minus{}x_4)\equal{}x_5x_6 \\
x_3x_4(1\minus{}x_5)\equal{}x_6x_1 \\
x_4x_5(1\minus{}x_6)\equal{}x_1x_2 \\
x_5x_6(1\minus{}x_1)\equal{}x_2x_3 \\
x_6x_1(1\minus{}x_2)\equal{}x_3x_4$
1978 IMO Longlists, 49
Let $A,B,C,D$ be four arbitrary distinct points in space.
$(a)$ Prove that using the segments $AB +CD, AC +BD$ and $AD +BC$, it is always possible to construct a triangle $T$ that is non-degenerate and has no obtuse angle.
$(b)$ What should these four points satisfy in order for the triangle $T$ to be right-angled?
1983 IMO Longlists, 75
Find the sum of the fiftieth powers of all sides and diagonals of a regular $100$-gon inscribed in a circle of radius $R.$
2025 Malaysian IMO Training Camp, 4
For each positive integer $k$, find all positive integer $n$ such that there exists a permutation $a_1,\ldots,a_n$ of $1,2,\ldots,n$ satisfying $$a_1a_2\ldots a_i\equiv i^k \pmod n$$ for each $1\le i\le n$.
[i](Proposed by Tan Rui Xuen and Ivan Chan Guan Yu)[/i]
2005 Tournament of Towns, 3
John and James wish to divide $25$ coins, of denominations $1, 2, 3, \ldots , 25$ kopeks. In each move, one of them chooses a coin, and the other player decides who must take this coin. John makes the initial choice of a coin, and in subsequent moves, the choice is made by the player having more kopeks at the time. In the event that there is a tie, the choice is made by the same player in the preceding move. After all the coins have been taken, the player with more kopeks wins. Which player has a winning strategy?
[i](5 points)[/i]
1962 Vietnam National Olympiad, 5
Solve the equation $ \sin^6x \plus{} \cos^6x \equal{} \frac{1}{4}$.
Geometry Mathley 2011-12, 12.4
Quadrilateral$ 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
2014 Greece National Olympiad, 2
Find all the integers $n$ for which $\frac{8n-25}{n+5}$ is cube of a rational number.
2024 Junior Balkan Team Selection Tests - Moldova, 5
Prove that a number of the form $80\dots01$ (there is at least 1 zero) can't be a perfect square.
1997 Tournament Of Towns, (554) 4
Two circles intersect at points $A$ and $B$. A common tangent touches the first circle at point $C$ and the second at point $D$. Let $\angle CBD > \angle CAD$. Let the line $CB$ intersect the second circle again at point $E$. Prove that $AD$ bisects the angle $\angle CAE$.
(P Kozhevnikov)
MBMT Team Rounds, 2020.20
Sam colors each tile in a 4 by 4 grid white or black. A coloring is called [i]rotationally symmetric[/i] if the grid can be rotated 90, 180, or 270 degrees to achieve the same pattern. Two colorings are called [i]rotationally distinct[/i] if neither can be rotated to match the other. How many rotationally distinct ways are there for Sam to color the grid such that the colorings are [i]not[/i] rotationally symmetric?
[i]Proposed by Gabriel Wu[/i]
2016 Moldova Team Selection Test, 10
Let $A_{1}A_{2} \cdots A_{14}$ be a regular $14-$gon. Prove that $A_{1}A_{3}\cap A_{5}A_{11}\cap A_{6}A_{9}\ne \emptyset$.
2005 Romania National Olympiad, 2
Let $G$ be a group with $m$ elements and let $H$ be a proper subgroup of $G$ with $n$ elements. For each $x\in G$ we denote $H^x = \{ xhx^{-1} \mid h \in H \}$ and we suppose that $H^x \cap H = \{e\}$, for all $x\in G - H$ (where by $e$ we denoted the neutral element of the group $G$).
a) Prove that $H^x=H^y$ if and only if $x^{-1}y \in H$;
b) Find the number of elements of the set $\bigcup_{x\in G} H^x$ as a function of $m$ and $n$.
[i]Calin Popescu[/i]
2001 China Team Selection Test, 2
Let \(L_3 = \{3\}\), \(L_n = \{3, 4, \ldots, h\}\) (where \(h > 3\)). For any given integer \(n \geq 3\), consider a graph \(G\) with \(n\) vertices that contains a Hamiltonian cycle \(C\) and has more than \(\frac{n^2}{4}\) edges. For which lengths \(l \in L_n\) must the graph \(G\) necessarily contain a cycle of length \(l\)?
MOAA Gunga Bowls, 2021.24
Freddy the Frog is situated at 1 on an infinitely long number line. On day $n$, where $n\ge 1$, Freddy can choose to hop 1 step to the right, stay where he is, or hop $k$ steps to the left, where $k$ is an integer at most $n+1$. After day 5, how many sequences of moves are there such that Freddy has landed on at least one negative number?
[i]Proposed by Andy Xu[/i]
2023 Harvard-MIT Mathematics Tournament, 19
Compute the number of ways to select $99$ cells in a $19 \times 19$ square grid such that no two selected cells share an edge or a vertex.
1983 Brazil National Olympiad, 2
An equilateral triangle $ABC$ has side a. A square is constructed on the outside of each side of the triangle. A right regular pyramid with sloping side $a$ is placed on each square. These pyramids are rotated about the sides of the triangle so that the apex of each pyramid comes to a common point above the triangle. Show that when this has been done, the other vertices of the bases of the pyramids (apart from the vertices of the triangle) form a regular hexagon.
2021 Purple Comet Problems, 11
Find the minimum possible value of |m -n|, where $m$ and $n$ are integers satisfying $m + n = mn - 2021$.
2011 Canadian Open Math Challenge, 8
A group of n friends wrote a math contest consisting of eight short-answer problem $S_1, S_2, S_3, S_4, S_5, S_6, S_7, S_8$, and four full-solution problems $F_1, F_2, F_3, F_4$. Each person in the group correctly solved exactly 11 of the 12 problems. We create an 8 x 4 table. Inside the square located in the $i$th row and $j$th column, we write down the number of people who correctly solved both problem $S_i$ and $F_j$. If the 32 entries in the table sum to 256, what is the value of n?
1972 Dutch Mathematical Olympiad, 3
$ABCD$ is a regular tetrahedron. The points $P,Q,R$ and $S$ lie outside this tetrahedron in such a way that $ABCP$, $ABDQ$, $ACDR$ and $BCDS$ are regular tetrahedra. Prove that the volume of the tetrahedron $PQRS$ is less than the sum of the volumes of $ABCP$,$ABDQ$,$ACDR$, $BCDS$ and $ABCD$.
2010 Brazil National Olympiad, 3
Find all pairs $(a, b)$ of positive integers such that
\[ 3^a = 2b^2 + 1. \]