Found problems: 85335
2016 Brazil National Olympiad, 1
Let $ABC$ be a triangle.
$r$ and $s$ are the angle bisectors of $\angle ABC$ and $\angle BCA$, respectively.
The points $E$ in $r$ and $D$ in $s$ are such that $AD \| BE$ and $AE \| CD$.
The lines $BD$ and $CE$ cut each other at $F$.
$I$ is the incenter of $ABC$.
Show that if $A,F,I$ are collinear, then $AB=AC$.
2018 PUMaC Algebra B, 7
For $k \in \left \{ 0, 1, \ldots, 9 \right \},$ let $\epsilon_k \in \left \{-1, 1 \right \}$. If the minimum possible value of $\sum_{i = 1}^9 \sum_{j = 0}^{i -1} \epsilon_i \epsilon_j 2^{i + j}$ is $m$, find $|m|$.
2020 ITAMO, 1
Let $\omega$ be a circle and let $A,B,C,D,E$ be five points on $\omega$ in this order. Define $F=BC\cap DE$, such that the points $F$ and $A$ are on opposite sides, with regard to the line $BE$ and the line $AE$ is tangent to the circumcircle of the triangle $BFE$.
a) Prove that the lines $AC$ and $DE$ are parallel
b) Prove that $AE=CD$
MBMT Team Rounds, 2015 F8 E5
Victor has $3$ piles of $3$ cards each. He draws all of the cards, but cannot draw a card until all the cards above it have been drawn. (For example, for his first card, Victor must draw the top card from one of the $3$ piles.) In how many orders can Victor draw the cards?
2009 Greece National Olympiad, 3
Let $ x,y,z$ be nonnegative real numbers such that $ x \plus{} y \plus{} z \equal{} 2$. Prove that $ x^{2}y^{2} \plus{} y^{2}z^{2} \plus{} z^{2}x^{2} \plus{} xyz\leq 1$. When does the equality occur?
2002 Tournament Of Towns, 1
Show that if the last digit of the number $x^2+xy+y^2$ is $0$ (where $x,y\in\mathbb{N}$ ) then last two digits are zero.
2005 All-Russian Olympiad, 2
Do there exist 12 rectangular parallelepipeds $P_1,\,P_2,\ldots,P_{12}$ with edges parallel to coordinate axes $OX,\,OY,\,OZ$ such that $P_i$ and $P_j$ have a common point iff $i\ne j\pm 1$ modulo 12?
2007 Harvard-MIT Mathematics Tournament, 30
$ABCD$ is a cyclic quadrilateral in which $AB=3$, $BC=5$, $CD=6$, and $AD=10$. $M$, $I$, and $T$ are the feet of the perpendiculars from $D$ to lines $AB$, $AC$, and $BC$ respectively. Determine the value of $MI/IT$.
2007 Pre-Preparation Course Examination, 11
Let $p \geq 3$ be a prime and $a_1,a_2,\cdots , a_{p-2}$ be a sequence of positive integers such that for every $k \in \{1,2,\cdots,p-2\}$ neither $a_k$ nor $a_k^k-1$ is divisible by $p$. Prove that product of some of members of this sequence is equivalent to $2$ modulo $p$.
2019 Junior Balkan Team Selection Tests - Romania, 3
A circle with center $O$ is internally tangent to two circles inside it at points $S$ and $T$. Suppose the two circles inside intersect at $M$ and $N$ with $N$ closer to $ST$. Show that $OM$ and $MN$ are perpendicular if and only if $S,N, T$ are collinear.
1985 ITAMO, 1
Let $x_1 = 97$, and for $n > 1$ let $x_n = \frac{n}{x_{n - 1}}$. Calculate the product $x_1 x_2 \dotsm x_8$.
2005 QEDMO 1st, 1 (Z4)
Prove that every integer can be written as sum of $5$ third powers of integers.
1996 Flanders Math Olympiad, 4
Consider a real poylnomial $p(x)=a_nx^n+...+a_1x+a_0$.
(a) If $\deg(p(x))>2$ prove that $\deg(p(x)) = 2 + deg(p(x+1)+p(x-1)-2p(x))$.
(b) Let $p(x)$ a polynomial for which there are real constants $r,s$ so that for all real $x$ we have \[ p(x+1)+p(x-1)-rp(x)-s=0 \]Prove $\deg(p(x))\le 2$.
(c) Show, in (b) that $s=0$ implies $a_2=0$.
2020 CMIMC Team, 4
Given $n=2020$, sort the $6$ values $$n^{n^2},\,\, 2^{2^{2^n}},\,\, n^{2^n},\,\, 2^{2^{n^2}},\,\, 2^{n^n},\,\,\text{and}\,\, 2^{n^{2^2}}$$ from [b]least[/b] to [b]greatest[/b]. Give your answer as a 6 digit permutation of the string "123456", where the number $i$ corresponds to the $i$-th expression in the list, from left to right.
1989 IMO Longlists, 45
Let $ (\log_2(x))^2 \minus{} 4 \cdot \log_2(x) \minus{} m^2 \minus{} 2m \minus{} 13 \equal{} 0$ be an equation in $ x.$ Prove:
[b](a)[/b] For any real value of $ m$ the equation has two distinct solutions.
[b](b)[/b] The product of the solutions of the equation does not depend on $ m.$
[b](c)[/b] One of the solutions of the equation is less than 1, while the other solution is greater than 1.
Find the minimum value of the larger solution and the maximum value of the smaller solution.
2019 BMT Spring, 3
There are several pairs of integers $ (a, b) $ satisfying $ a^2 - 4a + b^2 - 8b = 30 $. Find the sum of the sum of the coordinates of all such points.
2016 May Olympiad, 3
We say that a positive integer is [i]quad-divi[/i] if it is divisible by the sum of the squares of its digits, and also none of its digits is equal to zero.
a) Find a quad-divi number such that the sum of its digits is $24$.
b) Find a quad-divi number such that the sum of its digits is $1001$.
1972 Polish MO Finals, 3
Prove that there is a polynomial $P(x)$ with integer coefficients such that for all $x$ in the interval $\left[ \frac{1}{10}
, \frac{9}{10}\right]$ we have $$\left|P(x) -\frac12 \right| < \frac{ 1}{1000 }.$$
2013 South africa National Olympiad, 3
Let ABC be an acute-angled triangle and AD one of its altitudes (D on BC). The line through D parallel to AB is denoted by $l$, and t is the tangent to the circumcircle of ABC at A. Finally, let E be the intersection of $l$ and t. Show that CE and t are perpendicular to each other.
Kvant 2021, M2637
A table with three rows and 100 columns is given. Initially, in the left cell of each row there are $400\cdot 3^{100}$ chips. At one move, Petya marks some (but at least one) chips on the table, and then Vasya chooses one of the three rows. After that, all marked chips in the selected row are shifted to the right by a cell, and all marked chips in the other rows are removed from the table. Petya wins if one of the chips goes beyond the right edge of the table; Vasya wins if all the chips are removed. Who has a winning strategy?
[i]Proposed by P. Svyatokum, A. Khuzieva and D. Shabanov[/i]
2013 Thailand Mathematical Olympiad, 9
Let $ABCD$ be a convex quadrilateral, and let $M$ and$ N$ be midpoints of sides $AB$ and $CD$ respectively. Point $P$ is chosen on $CD$ so that $MP \perp CD$, and point $Q$ is chosen on $AB$ so that $NQ \perp AB$. Show that $AD \parallel BC$ if and only if $\frac{AB}{CD} =\frac{MP}{NQ}$ .
2024 China Girls Math Olympiad, 3
Find the smallest real $\lambda$, such that for any positive integers $n, a, b$, such that $n \nmid a+b$, there exists a positive integer $1 \leq k \leq n-1$, satisfying $$\{\frac{ak} {n}\}+\{\frac{bk} {n}\} \leq \lambda.$$
2006 IMC, 4
Let f be a rational function (i.e. the quotient of two real polynomials) and suppose that $f(n)$ is an integer for infinitely many integers n. Prove that f is a polynomial.
IV Soros Olympiad 1997 - 98 (Russia), 11.3
Solve the inequality
$$\sqrt{(x-2)^2(x-x^2)}<\sqrt{4x-1-(x^2-3x)^2}$$
2007 Indonesia TST, 3
On each vertex of a regular $ n\minus{}$gon there was a crow. Call this as initial configuration. At a signal, they all flew by and after a while, those $ n$ crows came back to the $ n\minus{}$gon, one crow for each vertex. Call this as final configuration. Determine all $ n$ such that: there are always three crows such that the triangle they formed in the initial configuration and the triangle they formed in the final configuration are both right-angled triangle.