Found problems: 85335
2019 SAFEST Olympiad, 4
Let $a_1, a_2, . . . , a_{2019}$ be any positive real numbers such that $\frac{1}{a_1 + 2019}+\frac{1}{a_2 + 2019}+ ... +\frac{1}{a_{2019} + 2019}=\frac{1}{2019}$.
Find the minimum value of $a_1a_2... a_{2019}$ and determine for which values of $a_1, a_2, . . . , a_{2019}$ this minimum occurs
2004 CHKMO, 2
Let $ABCDEF$ regular hexagon of side length $1$ and $O$ is its center. In addition to the sides of the hexagon, line segments from $O$ to the every vertex are drawn, making as total of $12$ unit segments. Find the number paths of length $2003$ along these segments that star at $O$ and terminate at $O$.
2011 Romanian Masters In Mathematics, 3
The cells of a square $2011 \times 2011$ array are labelled with the integers $1,2,\ldots, 2011^2$, in such a way that every label is used exactly once. We then identify the left-hand and right-hand edges, and then the top and bottom, in the normal way to form a torus (the surface of a doughnut).
Determine the largest positive integer $M$ such that, no matter which labelling we choose, there exist two neighbouring cells with the difference of their labels at least $M$.
(Cells with coordinates $(x,y)$ and $(x',y')$ are considered to be neighbours if $x=x'$ and $y-y'\equiv\pm1\pmod{2011}$, or if $y=y'$ and $x-x'\equiv\pm1\pmod{2011}$.)
[i](Romania) Dan Schwarz[/i]
Kvant 2020, M942
We divide the set $\{1,2,\cdots,2n\}$ into two disjoint sets : $\{a_1,a_2,\cdots,a_n\}$ and $\{b_1,b_2,\cdots,b_n\}$ such that :
$$a_1<a_2<\cdots<a_n\text{ and } b_1>b_2>\cdots>b_n.$$
Show that :
$$|a_1-b_1|+\cdots+|a_n-b_n|=n^2. $$
2017 USA Team Selection Test, 1
You are cheating at a trivia contest. For each question, you can peek at each of the $n > 1$ other contestants' guesses before writing down your own. For each question, after all guesses are submitted, the emcee announces the correct answer. A correct guess is worth $0$ points. An incorrect guess is worth $-2$ points for other contestants, but only $-1$ point for you, since you hacked the scoring system. After announcing the correct answer, the emcee proceeds to read the next question. Show that if you are leading by $2^{n - 1}$ points at any time, then you can surely win first place.
[i]Linus Hamilton[/i]
2021 Sharygin Geometry Olympiad, 15
Let $APBCQ$ be a cyclic pentagon. A point $M$ inside triangle $ABC$ is such that $\angle MAB = \angle MCA$, $\angle MAC = \angle MBA$ and $\angle PMB = \angle QMC = 90^\circ$. Prove that $AM$, $BP$, and $CQ$ concur.
[i]Anant Mudgal and Navilarekallu Tejaswi[/i]
2003 China Team Selection Test, 3
Let $A= \{a_1,a_2, \cdots, a_n \}$ and $B=\{b_1,b_2 \cdots, b_n \}$ be two positive integer sets and $|A \cap B|=1$. $C= \{ \text{all the 2-element subsets of A} \} \cup \{ \text{all the 2-element subsets of B} \}$. Function $f: A \cup B \to \{ 0, 1, 2, \cdots 2 C_n^2 \}$ is injective. For any $\{x,y\} \in C$, denote $|f(x)-f(y)|$ as the $\textsl{mark}$ of $\{x,y\}$. If $n \geq 6$, prove that at least two elements in $C$ have the same $\textsl{mark}$.
1977 IMO Longlists, 49
Find all pairs of integers $(p,q)$ for which all roots of the trinomials $x^2+px+q$ and $x^2+qx+p$ are integers.
1991 Tournament Of Towns, (285) 1
Prove that the product of the $99$ fractions
$$\frac{k^3-1}{k^3+1} \,\, , \,\,\,\,\,\, k=2,3,...,100$$
is greater than $2/3$.
(D. Fomin, Leningrad)
2003 Estonia National Olympiad, 3
In the rectangle $ABCD$ with $|AB|<2 |AD|$, let $E$ be the midpoint of $AB$ and $F$ a point on the chord $CE$ such that $\angle CFD = 90^o$. Prove that $FAD$ is an isosceles triangle.
2023 Malaysian IMO Training Camp, 1
For which $n\ge 3$ does there exist positive integers $a_1<a_2<\cdots <a_n$, such that: $$a_n=a_1+...+a_{n-1}, \hspace{0.5cm} \frac{1}{a_1}=\frac{1}{a_2}+...+\frac{1}{a_n}$$ are both true?
[i]Proposed by Ivan Chan Kai Chin[/i]
1990 AMC 12/AHSME, 22
If the six solutions of $x^6=-64$ are written in the form $a+bi$, where $a$ and $b$ are real, then the product of those solutions with $a>0$ is
$\text{(A)} \ -2 \qquad \text{(B)} \ 0 \qquad \text{(C)} \ 2i \qquad \text{(D)} \ 4 \qquad \text{(E)} \ 16$
1957 Miklós Schweitzer, 9
[b]9.[/b] Find all pairs of linear polynomials $f(x)$, $g(x)$ with integer coefficients for which there exist two polynomials $u(x)$, $v(x)$ with integer coefficients such that $f(x)u(x)+g(x)v(x)=1$. [b](A. 8)[/b]
2007 Germany Team Selection Test, 2
Let $ n, k \in \mathbb{N}$ with $ 1 \leq k \leq \frac {n}{2} - 1.$ There are $ n$ points given on a circle. Arbitrarily we select $ nk + 1$ chords among the points on the circle. Prove that of these chords there are at least $ k + 1$ chords which pairwise do not have a point in common.
2017 HMNT, 10
Five equally skilled tennis players named Allen, Bob, Catheryn, David, and Evan play in a round robin tournament, such that each pair of people play exactly once, and there are no ties. In each of the ten games, the two players both have a 50% chance of winning, and the results of the games are independent. Compute the probability that there exist four distinct players $P_1$, $P_2$, $P_3$, $P_4$ such that $P_i$ beats $P_{i+1}$ for $i=1, 2, 3, 4$. (We denote $P_5=P_1$).
2013 Today's Calculation Of Integral, 893
Find the minimum value of $f(x)=\int_0^{\frac{\pi}{4}} |\tan t-x|dt.$
2019 India PRMO, 6
Let $ABC$ be a triangle such that $AB=AC$. Suppose the tangent to the circumcircle of ABC at B is perpendicular to AC. Find angle ABC measured in degrees
2025 Macedonian TST, Problem 6
Let $n>2$ be an even integer, and let $V$ be an arbitrary set of $8$ distinct integers. Define
\[
E(V,n)
\;=\;
\bigl\{(u,v)\in V\times V : u < v,\ u+v = n^k\text{ for some }k\in\mathbb{N}\bigr\}.
\]
For each even $n>2$, determine the maximum possible size of the set $E(V,n)$.
2021-2022 OMMC, 23
A $39$-tuple of real numbers $(x_1,x_2,\ldots x_{39})$ satisfies
\[2\sum_{i=1}^{39} \sin(x_i) = \sum_{i=1}^{39} \cos(x_i) = -34.\]
The ratio between the maximum of $\cos(x_1)$ and the maximum of $\sin(x_1)$ over all tuples $(x_1,x_2,\ldots x_{39})$ satisfying the condition is $\tfrac ab$ for coprime positive integers $a$, $b$ (these maxima aren't necessarily achieved using the same tuple of real numbers). Find $a + b$.
[i]Proposed by Evan Chang[/i]
2007 Pan African, 2
Let $A$, $B$ and $C$ be three fixed points, not on the same line. Consider all triangles $AB'C'$ where $B'$ moves on a given straight line (not containing $A$), and $C'$ is determined such that $\angle B'=\angle B$ and $\angle C'=\angle C$. Find the locus of $C'$.
2006 Tournament of Towns, 4
Quadrilateral $ABCD$ is a cyclic, $AB = AD$. Points $M$ and $N$ are chosen on sides $BC$ and $CD$ respectfully so that $\angle MAN =1/2 (\angle BAD)$. Prove that $MN = BM + ND$.
[i](5 points)[/i]
2017 South East Mathematical Olympiad, 4
For any positive integer $n$, let $D_n$ denote the set of all positive divisors of $n$, and let $f_i(n)$ denote the size of the set
$$F_i(n) = \{a \in D_n | a \equiv i \pmod{4} \}$$where $i = 0, 1, 2, 3$.
Determine the smallest positive integer $m$ such that $f_0(m) + f_1(m) - f_2(m) - f_3(m) = 2017$.
2015 CCA Math Bonanza, L1.2
Let $ABCDEF$ be a regular hexagon with side length $2$. Calculate the area of $ABDE$.
[i]2015 CCA Math Bonanza Lightning Round #1.2[/i]
1995 South africa National Olympiad, 2
Find all pairs $(m,n)$ of natural numbers with $m<n$ such that $m^2+1$ is a multiple of $n$ and $n^2+1$ is a multiple of $m$.
2003 China Team Selection Test, 2
Denote by $\left(ABC\right)$ the circumcircle of a triangle $ABC$.
Let $ABC$ be an isosceles right-angled triangle with $AB=AC=1$ and $\measuredangle CAB=90^{\circ}$. Let $D$ be the midpoint of the side $BC$, and let $E$ and $F$ be two points on the side $BC$.
Let $M$ be the point of intersection of the circles $\left(ADE\right)$ and $\left(ABF\right)$ (apart from $A$).
Let $N$ be the point of intersection of the line $AF$ and the circle $\left(ACE\right)$ (apart from $A$).
Let $P$ be the point of intersection of the line $AD$ and the circle $\left(AMN\right)$.
Find the length of $AP$.