Found problems: 85335
2024 Thailand October Camp, 3
Recall that for an arbitrary prime $p$, we define a [b]primitive root[/b] modulo $p$ as an integer $r$ for which the least positive integer $v$ such that $r^{v}\equiv 1\pmod{p}$ is $p-1$.\\
Prove or disprove the following statement:
[center]
For every prime $p>2023$, there exists positive integers $1\leqslant a<b<c<p$\\ such that $a,b$ and $c$ are primitive roots modulo $p$ but $abc$ is not a primitive root modulo $p$.
[/center]
1992 AMC 8, 13
Five test scores have a mean (average score) of $90$, a median (middle score) of $91$ and a mode (most frequent score) of $94$. The sum of the two lowest test scores is
$\text{(A)}\ 170 \qquad \text{(B)}\ 171 \qquad \text{(C)}\ 176 \qquad \text{(D)}\ 177 \qquad \text{(E)}\ \text{not determined by the information given}$
2012 NIMO Summer Contest, 4
The degree measures of the angles of nondegenerate hexagon $ABCDEF$ are integers that form a non-constant arithmetic sequence in some order, and $\angle A$ is the smallest angle of the (not necessarily convex) hexagon. Compute the sum of all possible degree measures of $\angle A$.
[i]Proposed by Lewis Chen[/i]
1996 Korea National Olympiad, 6
Find the minimum value of $k$ such that there exists two sequence ${a_i},{b_i}$ for $i=1,2,\cdots ,k$ that satisfies the following conditions.
(i) For all $i=1,2,\cdots ,k,$ $a_i,b_i$ is the element of $S=\{1996^n|n=0,1,2,\cdots\}.$
(ii) For all $i=1,2,\cdots, k, a_i\ne b_i.$
(iii) For all $i=1,2,\cdots, k, a_i\le a_{i+1}$ and $b_i\le b_{i+1}.$
(iv) $\sum_{i=1}^{k} a_i=\sum_{i=1}^{k} b_i.$
2003 IMO Shortlist, 1
Let $A$ be a $101$-element subset of the set $S=\{1,2,\ldots,1000000\}$. Prove that there exist numbers $t_1$, $t_2, \ldots, t_{100}$ in $S$ such that the sets \[ A_j=\{x+t_j\mid x\in A\},\qquad j=1,2,\ldots,100 \] are pairwise disjoint.
2014 Balkan MO Shortlist, G1
Let $ABC$ be an isosceles triangle, in which $AB=AC$ , and let $M$ and $N$ be two points on the sides $BC$ and $AC$, respectively such that $\angle BAM = \angle MNC$. Suppose that the lines $MN$ and $AB$ intersects at $P$. Prove that the bisectors of the angles $\angle BAM$ and $\angle BPM$ intersects at a point lying on the line $BC$
2019 Purple Comet Problems, 13
There are relatively prime positive integers $m$ and $n$ so that the parabola with equation $y = 4x^2$ is tangent to the parabola with equation $x = y^2 + \frac{m}{n}$ . Find $m + n$.
1989 IMO Shortlist, 10
Let $ g: \mathbb{C} \rightarrow \mathbb{C}$, $ \omega \in \mathbb{C}$, $ a \in \mathbb{C}$, $ \omega^3 \equal{} 1$, and $ \omega \ne 1$. Show that there is one and only one function $ f: \mathbb{C} \rightarrow \mathbb{C}$ such that
\[ f(z) \plus{} f(\omega z \plus{} a) \equal{} g(z),z\in \mathbb{C}
\]
2021 Durer Math Competition (First Round), 1
Albrecht is travelling in his car on the motorway at a constant speed. The journey is very long so Marvin who is sitting next to Albrecht gets bored and decides to calculate the speed of the car. He was a bit careless but he noted that at noon they passed milestone $XY$ (where $X$ and $Y$ are digits), at $12:42$ milestone $YX$ and at $1$pm they arrived at milestone $X0Y$. What did Marvin deduce, what is the speed of the car?
2004 China Western Mathematical Olympiad, 4
Suppose that $ a$, $ b$, $ c$ are positive real numbers, prove that
\[ 1 < \frac {a}{\sqrt {a^{2} \plus{} b^{2}}} \plus{} \frac {b}{\sqrt {b^{2} \plus{} c^{2}}} \plus{} \frac {c}{\sqrt {c^{2} \plus{} a^{2}}}\leq\frac {3\sqrt {2}}{2}
\]
1998 USAMTS Problems, 3
Let $f$ be a polynomial of degree $98$, such that $f (k) =\frac{1}{k}$ for $k=1,2,3,\ldots,99$. Determine $f(100)$.
1960 Putnam, A7
Let $N(n)$ denote the smallest positive integer $N$ such that $x^N =e$ for every element $x$ of the symmetric group $S_n$, where $e$ denotes the identity permutation. Prove that if $n>1,$
$$\frac{N(n)}{N(n-1)} =\begin{cases} p \;\text{if}\; n\; \text{is a power of a prime } p\\
1\; \text{otherwise}.
\end{cases}$$
2017 Sharygin Geometry Olympiad, 1
If two circles intersect at $A,B$ and common tangents of them intesrsect circles at $C,D$if $O_a$is circumcentre of $ACD$ and $O_b$ is circumcentre of $BCD$ prove $AB$ intersects $O_aO_b$ at its midpoint
1980 Bundeswettbewerb Mathematik, 2
In a triangle $ABC$, the bisectors of angles $A$ and $B$ meet the opposite sides of the triangle at points $D$ and $E$, respectively. A point $P$ is arbitrarily chosen on the line $DE$. Prove that the distance of $P$ from line $AB$ equals the sum or the difference of the distances of $P$ from lines $AC$ and $BC$.
2023 VN Math Olympiad For High School Students, Problem 7
Given a triangle $ABC$ with symmedians $BE,CF(E,F$ are on the sides $CA,AB,$ respectively$)$ intersecting at [i]Lemoine[/i] point $L.$ Prove that: $AB=AC$ in each case:
a) $LB=LC.$
b) $BE=CF.$
2008 Postal Coaching, 6
Consider the set $A = \{1, 2, 3, ..., 2008\}$. We say that a set is of [i]type[/i] $r, r \in \{0, 1, 2\}$, if that set is a nonempty subset of $A$ and the sum of its elements gives the remainder $r$ when divided by $3$. Denote by $X_r, r \in \{0, 1, 2\}$ the class of sets of type $r$. Determine which of the classes $X_r, r \in \{0, 1, 2\}$, is the largest.
2020 IMO Shortlist, A1
[i]Version 1[/i]. Let $n$ be a positive integer, and set $N=2^{n}$. Determine the smallest real number $a_{n}$ such that, for all real $x$,
\[
\sqrt[N]{\frac{x^{2 N}+1}{2}} \leqslant a_{n}(x-1)^{2}+x .
\]
[i]Version 2[/i]. For every positive integer $N$, determine the smallest real number $b_{N}$ such that, for all real $x$,
\[
\sqrt[N]{\frac{x^{2 N}+1}{2}} \leqslant b_{N}(x-1)^{2}+x .
\]
2025 Kyiv City MO Round 2, Problem 4
Let \( H \) be the orthocenter, and \( O \) be the circumcenter of \( \triangle ABC \). The line \( AH \) intersects the circumcircle of \( \triangle ABC \) at point \( N \) for the second time. The circumcircle of \( \triangle BOC \), with center at point \( Q \), intersects the line \( OH \) at point \( X \) for the second time. Prove that the points \( O, Q, N, X \) lie on the same circle.
[i]Proposed by Matthew Kurskyi[/i]
2009 International Zhautykov Olympiad, 3
In a checked $ 17\times 17$ table, $ n$ squares are colored in black. We call a line any of rows, columns, or any of two diagonals of the table. In one step, if at least $ 6$ of the squares in some line are black, then one can paint all the squares of this line in black.
Find the minimal value of $ n$ such that for some initial arrangement of $ n$ black squares one can paint all squares of the table in black in some steps.
1969 IMO Shortlist, 10
$(BUL 4)$ Let $M$ be the point inside the right-angled triangle $ABC (\angle C = 90^{\circ})$ such that $\angle MAB = \angle MBC = \angle MCA =\phi.$ Let $\Psi$ be the acute angle between the medians of $AC$ and $BC.$ Prove that $\frac{\sin(\phi+\Psi)}{\sin(\phi-\Psi)}= 5.$
2013 Costa Rica - Final Round, N1
Find all triples $(a, b, p)$ of positive integers, where $p$ is a prime number, such that $a^p - b^p = 2013$.
2018 IMAR Test, 4
Prove that every non-negative integer $n$ is expressible in the form $n=t^2+u^2+v^2+w^2$, where $t,u,v,w$ are integers such that $t+u+v+w$ is a perfect square.
[i]* * *[/i]
1986 Poland - Second Round, 6
In the triangle $ ABC $, the point $ A' $ on the side $ BC $, the point $ B' $ on the side $ AC $, the point $ C' $ on the side $ AB $ are chosen so that the straight lines $ AA' $, $ CC' $ intersect at one point, i.e. equivalently $ |BA'| \cdot |CB'| \cdot |AC'| = |CA'| \cdot |AB'| \cdot |BC'| $. Prove that the area of triangle $ A'B'C' $ is not greater than $ 1/4 $ of the area of triangle $ ABC $.
2008 Princeton University Math Competition, 7
The graphs of the following equations divide the $xy$ plane into some number of regions.
$4 + (x + 2)y =x^2$
$(x + 2)^2 + y^2 =16$
Find the area of the second smallest region.
1969 Miklós Schweitzer, 9
In $ n$-dimensional Euclidean space, the union of any set of closed balls (of positive radii) is measurable in the sense of Lebesgue.
[i]A. Csaszar[/i]