Found problems: 85335
2021 Baltic Way, 13
Let $D$ be the foot of the $A$-altitude of an acute triangle $ABC$. The internal bisector of the angle $DAC$ intersects $BC$ at $K$. Let $L$ be the projection of $K$ onto $AC$. Let $M$ be the intersection point of $BL$ and $AD$. Let $P$ be the intersection point of $MC$ and $DL$. Prove that $PK \perp AB$.
2016 IFYM, Sozopol, 4
Circle $k$ passes through $A$ and intersects the sides of $\Delta ABC$ in $P,Q$, and $L$. Prove that:
$\frac{S_{PQL}}{S_{ABC}}\leq \frac{1}{4} (\frac{PL}{AQ})^2$.
LMT Guts Rounds, 2020 F10
$2020$ magicians are divided into groups of $2$ for the Lexington Magic Tournament. After every $5$ days, which is the duration of one match, teams are rearranged so no $2$ people are ever on the same team. If the longest tournament is $n$ days long, what is the value of $n?$
[i]Proposed by Ephram Chun[/i]
1951 AMC 12/AHSME, 21
Given: $ x > 0, y > 0, x > y$ and $ z\not \equal{} 0$. The inequality which is not always correct is:
$ \textbf{(A)}\ x \plus{} z > y \plus{} z \qquad\textbf{(B)}\ x \minus{} z > y \minus{} z \qquad\textbf{(C)}\ xz > yz$
$ \textbf{(D)}\ \frac {x}{z^2} > \frac {y}{z^2} \qquad\textbf{(E)}\ xz^2 > yz^2$
1950 Putnam, A6
Each coefficient $a_n$ of the power series \[a_0 + a_1 x + a_2 x^2 + a_3 x^3 + \cdots = f(x)\] has either the value of $1$ or the value $0.$ Prove the easier of the two assertions:
(i) If $f(0.5)$ is a rational number, $f(x)$ is a rational function.
(ii) If $f(0.5)$ is not a rational number, $f(x)$ is not a rational function.
2023 Thailand Mathematical Olympiad, 10
To celebrate the 20th Thailand Mathematical Olympiad (TMO), Ratchasima Witthayalai School put up flags around the Thao Suranari Monument so that
[list=i]
[*] Each flag is painted in exactly one color, and at least $2$ distinct colors are used.
[*] The number of flags are odd.
[*] Every flags are on a regular polygon such that each vertex has one flag.
[*] Every flags with the same color are on a regular polygon.
[/list]
Prove that there are at least $3$ colors with the same amount of flags.
2008 F = Ma, 13
A mass is attached to the wall by a spring of constant $k$. When the spring is at its natural length, the mass is given a certain initial velocity, resulting in oscillations of amplitude $A$. If the spring is replaced by a spring of constant $2k$, and the mass is given the same initial velocity, what is the amplitude of the resulting oscillation?
(a) $\frac{1}{2}A$
(b) $\frac{1}{\sqrt{2}}A$
(c) $\sqrt{2}A$
(d) $2A$
(e) $4A$
2018 Vietnam National Olympiad, 5
For two positive integers $n$ and $d$, let $S_n(d)$ be the set of all ordered $d$-tuples $(x_1,x_2,\dots ,x_d)$ that satisfy all of the following conditions:
i. $x_i\in \{1,2,\dots ,n\}$ for every $i\in\{1,2,\dots ,d\}$;
ii. $x_i\ne x_{i+1}$ for every $i\in\{1,2,\dots ,d-1\}$;
iii. There does not exist $i,j,k,l\in\{1,2,\dots ,d\}$ such that $i<j<k<l$ and $x_i=x_k,\, x_j=x_l$;
a. Compute $|S_3(5)|$
b. Prove that $|S_n(d)|>0$ if and only if $d\leq 2n-1$.
2025 Korea - Final Round, P1
Sequence $a_1, a_2, a_3, \cdots$ satisfies the following condition.
[b](Condition)[/b] For all positive integer $n$, $\sum_{k=1}^{n}\frac{1}{2}\left(1 - (-1)^{\left[\frac{n}{k}\right]}\right)a_k=1$ holds.
For a positive integer $m = 1001 \cdot 2^{2025}$, compute $a_m$.
PEN S Problems, 22
The decimal expression of the natural number $a$ consists of $n$ digits, while that of $a^3$ consists of $m$ digits. Can $n+m$ be equal to $2001$?
1971 IMO, 1
Let \[ E_n=(a_1-a_2)(a_1-a_3)\ldots(a_1-a_n)+(a_2-a_1)(a_2-a_3)\ldots(a_2-a_n)+\ldots+(a_n-a_1)(a_n-a_2)\ldots(a_n-a_{n-1}). \] Let $S_n$ be the proposition that $E_n\ge0$ for all real $a_i$. Prove that $S_n$ is true for $n=3$ and $5$, but for no other $n>2$.
2005 Argentina National Olympiad, 4
We will say that a positive integer is a [i]winner [/i] if it can be written as the sum of a perfect square plus a perfect cube. For example, $33$ is a winner because $33=5^2+2^3$ .
Gabriel chooses two positive integers, r and s, and Germán must find $2005$ positive integers $n$ such that for each $n$, the numbers $r+n$ and $s+n$ are winners.
Prove that Germán can always achieve his goal.
1993 Poland - First Round, 12
Prove that the sums of the opposite dihedral angles of a tetrahedron are equal if and only if the sums of the opposite edges of this tetrahedron are equal.
2025 Caucasus Mathematical Olympiad, 1
Anya and Vanya’s houses are located on the straight road. The distance between their houses is divided by a shop and a school into three equal parts. If Anya and Vanya leave their houses at the same time and walk towards each other, they will meet near the shop. If Anya rides a scooter, then her speed will increase by $150\,\text{m/min}$, and they will meet near the school. Find Vanya’s speed of walking.
2008 Oral Moscow Geometry Olympiad, 5
There are two shawls, one in the shape of a square, the other in the shape of a regular triangle, and their perimeters are the same. Is there a polyhedron that can be completely pasted over with these two shawls without overlap (shawls can be bent, but not cut)?
(S. Markelov).
2011 Spain Mathematical Olympiad, 2
Each rational number is painted either white or red. Call such a coloring of the rationals [i]sanferminera[/i] if for any distinct rationals numbers $x$ and $y$ satisfying one of the following three conditions: [list=1][*]$xy=1$,
[*]$x+y=0$,
[*]$x+y=1$,[/list]we have $x$ and $y$ painted different colors. How many sanferminera colorings are there?
1999 Italy TST, 1
Prove that for any prime number $p$ the equation $2^p+3^p=a^n$ has no solution $(a,n)$ in integers greater than $1$.
2004 CentroAmerican, 2
Let $ABCD$ be a trapezium such that $AB||CD$ and $AB+CD=AD$. Let $P$ be the point on $AD$ such that $AP=AB$ and $PD=CD$.
$a)$ Prove that $\angle BPC=90^{\circ}$.
$b)$ $Q$ is the midpoint of $BC$ and $R$ is the point of intersection between the line $AD$ and the circle passing through the points $B,A$ and $Q$. Show that the points $B,P,R$ and $C$ are concyclic.
1977 Chisinau City MO, 137
Determine the angles of a triangle in which the median, bisector and altitude, drawn from one vertex, divide this angle into four equal parts.
2020 Turkey EGMO TST, 3
There are $33!$ empty boxes labeled from $1$ to $33!$. In every move, we find the empty box with the smallest label, then we transfer $1$ ball from every box with a smaller label and we add an additional $1$ ball to that box. Find the smallest labeled non-empty box and the number of the balls in it after $33!$ moves.
1999 Slovenia National Olympiad, Problem 2
Consider the polynomial $p(x)=x^{1999}+2x^{1998}+3x^{1997}+\ldots+2000$. Find a nonzero polynomial whose roots are the reciprocal values of the roots of $p(x)$.
2024 LMT Fall, 17
Suppose $x$, $y$, $z$ are pairwise distinct real numbers satisfying
\[
x^2+3y =y^2 +3z = z^2+3x.
\]Find $(x+y)(y+z)(z+x)$.
2001 China Team Selection Test, 2
If the sum of all positive divisors (including itself) of a positive integer $n$ is $2n$, then $n$ is called a perfect number. For example, the sum of the positive divisors of 6 is $1 + 2 + 3 + 6 = 2 \times 6$, hence 6 is a perfect number.
Prove: There does not exist a perfect number of the form $p^a q^b r^c$, where $a, b, c$ are positive integers, and $p, q, r$ are odd primes.
2005 Switzerland - Final Round, 4
Determine all sets $M$ of natural numbers such that for every two (not necessarily different) elements $a, b$ from $M$ , $$\frac{a + b}{gcd(a, b)}$$ lies in $M$.
2009 Paraguay Mathematical Olympiad, 5
In a triangle $ABC$, let $I$ be its incenter. The distance from $I$ to the segment $BC$ is $4 cm$ and the distance from that point to vertex $B$ is $12 cm$. Let $D$ be a point in the plane region between segments $AB$ and $BC$ such that $D$ is the center of a circumference that is tangent to lines $AB$ and $BC$ and passes through $I$. Find all possible values of the length $BD$.