Found problems: 85335
1993 AIME Problems, 2
During a recent campaign for office, a candidate made a tour of a country which we assume lies in a plane. On the first day of the tour he went east, on the second day he went north, on the third day west, on the fourth day south, on the fifth day east, etc. If the candidate went $n^2/2$ miles on the $n^{\text{th}}$ day of this tour, how many miles was he from his starting point at the end of the $40^{\text{th}}$ day?
1989 National High School Mathematics League, 3
For any function $f(x)$, in the same rectangular coordinates, figures of function $y=f(x-1)$ and $y=f(-x+1)$
$\text{(A)}$ are symmetrical about $x$-axis
$\text{(B)}$ are symmetrical about line $x=1$
$\text{(C)}$ are symmetrical about line $x=-1$
$\text{(D)}$ are symmetrical about $y$-axis
2006 China Second Round Olympiad, 9
Suppose points $F_1, F_2$ are the left and right foci of the ellipse $\frac{x^2}{16}+\frac{y^2}{4}=1$ respectively, and point $P$ is on line $l:$, $x-\sqrt{3} y+8+2\sqrt{3}=0$. Find the value of ratio $\frac{|PF_1|}{|PF_2|}$ when $\angle F_1PF_2$ reaches its maximum value.
2018 Belarusian National Olympiad, 10.4
Some cells of a checkered plane are marked so that figure $A$ formed by marked cells satisfies the following condition:$1)$ any cell of the figure $A$ has exactly two adjacent cells of $A$; and $2)$ the figure $A$ can be divided into isosceles trapezoids of area $2$ with vertices at the grid nodes (and acute angles of trapezoids are equal to $45$) . Prove that the number of marked cells is divisible by $8$.
2018 Harvard-MIT Mathematics Tournament, 1
Consider a $2\times 3$ grid where each entry is either $0$, $1$, or $2$. For how many such grids is the sum of the numbers in every row and in every column a multiple of $3$? One valid grid is shown below:
$$\begin{bmatrix} 1 & 2 & 0 \\ 2 & 1 & 0 \end{bmatrix}$$
2017 NIMO Summer Contest, 14
Let $x, y, z$ be real numbers such that $x+y+z=-2$ and
\[\begin{aligned}
& (x^2+xy+y^2)(y^2+yz+z^2) \\
&+ (y^2+yz+z^2)(z^2+zx+x^2) \\
&+ (z^2+zx+x^2)(x^2+xy+y^2) \\
& = 625+ \tfrac34(xy+yz+zx)^2. \end{aligned}\]
Compute $|xy+yz+zx|$.
[i]Proposed by Michael Tang[/i]
2022 Iranian Geometry Olympiad, 1
Four points $A$, $B$, $C$ and $D$ lie on a circle $\omega$ such that $AB=BC=CD$. The tangent line to $\omega$ at point $C$ intersects the tangent line to $\omega$ at $A$ and the line $AD$ at $K$ and $L$. The circle $\omega$ and the circumcircle of triangle $KLA$ intersect again at $M$. Prove that $MA=ML$.
[i]Proposed by Mahdi Etesamifard[/i]
2019 Argentina National Olympiad Level 2, 2
A $7 \times 7$ grid is given. Julián colors $29$ cells black. Pilar must then place an $L$-shaped piece, covering exactly three cells (oriented in any direction, as shown in the figure). Pilar wins if all three cells covered by the $L$-shaped piece are black. Can Julián color the grid in such a way that it is impossible for Pilar to win?
[asy]
size(1.5cm);
draw((0,1)--(1,1)--(1,2)--(0,2)--(0,1)--(0,0)--(1,0)--(2,0)--(2,1)--(1,1)--(1,0));
[/asy]
Indonesia MO Shortlist - geometry, g9
Given a triangle $ABC$, the points $D$, $E$, and $F$ lie on the sides $BC$, $CA$, and $AB$, respectively, are such that
$$DC + CE = EA + AF = FB + BD.$$ Prove that $$DE + EF + FD \ge \frac12 (AB + BC + CA).$$
Cono Sur Shortlist - geometry, 2012.G5
Let $ABC$ be an acute triangle, and let $H_A$, $H_B$, and $H_C$ be the feet of the altitudes relative to vertices $A$, $B$, and $C$, respectively. Define $I_A$, $I_B$, and $I_C$ as the incenters of triangles $AH_B H_C$, $BH_C H_A$, and $CH_A H_B$, respectively. Let $T_A$, $T_B$, and $T_C$ be the intersection of the incircle of triangle $ABC$ with $BC$, $CA$, and $AB$, respectively. Prove that the triangles $I_A I_B I_C$ and $T_A T_B T_C$ are congruent.
2009 Middle European Mathematical Olympiad, 9
Let $ ABCD$ be a parallelogram with $ \angle BAD \equal{} 60$ and denote by $ E$ the intersection of its diagonals. The circumcircle of triangle $ ACD$ meets the line $ BA$ at $ K \ne A$, the line $ BD$ at $ P \ne D$ and the line $ BC$ at $ L\ne C$. The line $ EP$ intersects the circumcircle of triangle $ CEL$ at points $ E$ and $ M$. Prove that triangles $ KLM$ and $ CAP$ are congruent.
2023 Indonesia MO, 5
Let $a$ and $b$ be positive integers such that $\text{gcd}(a, b) + \text{lcm}(a, b)$ is a multiple of $a+1$. If $b \le a$, show that $b$ is a perfect square.
Russian TST 2014, P1
Let $x,y,z$ be positive real numbers. Prove that \[\frac{x}{y}+\frac{y}{z}+\frac{z}{x}\geqslant\frac{z(x+y)}{y(y+z)}+\frac{x(y+z)}{z(z+x)}+\frac{y(z+x)}{x(x+y)}.\]
2008 Kyiv Mathematical Festival, 4
Let $ K,L,M$ and $ N$ be the midpoints of sides $ AB,$ $ BC,$ $ CD$ and $ AD$ of the convex quadrangle $ ABCD.$ Is it possible that points $ A,B,L,M,D$ lie on the same circle and points $ K,B,C,D,N$ lie on the same circle?
2017 Moscow Mathematical Olympiad, 9
There are $80$ peoples, one of them is murderer, and other one is witness of crime. Every day detective
interrogates some peoples from this group. Witness will says about crime only if murderer will not be in interrogatory with him. It is enough $12$ days to find murderer ?
2021 BMT, 12
Let $a$, $b$, and $c$ be the solutions of the equation $$x^3 - 3 \cdot 2021^2x = 2 \cdot 20213.$$ Compute $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}.$
2009 ELMO Problems, 5
Let $ABCDEFG$ be a regular heptagon with center $O$. Let $M$ be the centroid of $\triangle ABD$. Prove that $\cos^2(\angle GOM)$ is rational and determine its value.
[i]Evan o'Dorney[/i]
2016 Baltic Way, 16
In triangle $ABC,$ the points $D$ and $E$ are the intersections of the angular bisectors from $C$ and $B$ with the sides $AB$ and $AC,$ respectively. Points $F$ and $G$ on the extensions of $AB$ and $AC$ beyond $B$ and $C,$ respectively, satisfy $BF = CG = BC.$ Prove that $F G \parallel DE.$
1993 All-Russian Olympiad Regional Round, 9.2
Find the largest natural number which cannot be turned into a multiple of $11$ by reordering its (decimal) digits.
1996 Miklós Schweitzer, 6
Let $\{a_n\}$ be a bounded real sequence.
(a) Prove that if X is a positive-measure subset of $\mathbb R$, then for almost all $x\in X$, there exist a subsequence $\{y_n\}$ of X such that $$\sum_{n=1}^\infty (n(y_n-x)-a_n)=1$$
(b) construct an unbounded sequence $\{a_n\}$ for which the above equation is also true.
1996 All-Russian Olympiad, 6
In isosceles triangle $ABC$ ($AB = BC$) one draws the angle bisector $CD$. The perpendicular to $CD$ through the center of the circumcircle of $ABC$ intersects $BC$ at $E$. The parallel to $CD$ through $E$ meets $AB$ at $F$. Show that $BE$ = $FD$.
[i]M. Sonkin[/i]
2009 China Team Selection Test, 2
In convex quadrilateral $ ABCD$, $ CB,DA$ are external angle bisectors of $ \angle DCA,\angle CDB$, respectively. Points $ E,F$ lie on the rays $ AC,BD$ respectively such that $ CEFD$ is cyclic quadrilateral. Point $ P$ lie in the plane of quadrilateral $ ABCD$ such that $ DA,CB$ are external angle bisectors of $ \angle PDE,\angle PCF$ respectively. $ AD$ intersects $ BC$ at $ Q.$ Prove that $ P$ lies on $ AB$ if and only if $ Q$ lies on segment $ EF$.
2004 Harvard-MIT Mathematics Tournament, 10
A [i]lattice point[/i] is a point whose coordinates are both integers. Suppose Johann walks in a line from the point $(0, 2004)$ to a random lattice point in the interior (not on the boundary) of the square with vertices $(0, 0)$, $(0, 99)$, $(99,99)$, $(99, 0)$. What is the probability that his path, including the endpoints, contains an even number of lattice points?
2021 Princeton University Math Competition, 2
Let $k \in Z_{>0}$ be the smallest positive integer with the property that $k\frac{gcd(x,y)gcd(y,z)}{lcm (x,y^2,z)}$ is a positive integer for all values $1 \le x \le y \le z \le 121$. If k' is the number of divisors of $k$, find the number of divisors of $k'$.
2011 Saudi Arabia Pre-TST, 3.1
Let $n$ be a positive integer such that $2011^{2011}$ divides $n!$. Prove that $2011^{2012} $divides $n!$ .