Found problems: 85335
2004 Pan African, 3
Let $ABCD$ be a cyclic quadrilateral such that $AB$ is a diameter of it's circumcircle. Suppose that $AB$ and $CD$ intersect at $I$, $AD$ and $BC$ at $J$, $AC$ and $BD$ at $K$, and let $N$ be a point on $AB$. Show that $IK$ is perpendicular to $JN$ if and only if $N$ is the midpoint of $AB$.
2002 Romania Team Selection Test, 1
Let $ABCDE$ be a cyclic pentagon inscribed in a circle of centre $O$ which has angles $\angle B=120^{\circ},\angle C=120^{\circ},$ $\angle D=130^{\circ},\angle E=100^{\circ}$. Show that the diagonals $BD$ and $CE$ meet at a point belonging to the diameter $AO$.
[i]Dinu Șerbănescu[/i]
2020 ISI Entrance Examination, 4
Let a real-valued sequence $\{x_n\}_{n\geqslant 1}$ be such that $$\lim_{n\to\infty}nx_n=0$$ Find all possible real values of $t$ such that $\lim_{n\to\infty}x_n\big(\log n\big)^t=0$ .
2020 Durer Math Competition Finals, 16
Dora has $8$ rods with lengths $1, 2, 3, 4, 5, 6, 7$ and $8$ cm. Dora chooses $4$ of the rods and uses them to assemble a trapezoid (the $4$ chosen rods must be the $4$ sides). How many different trapezoids can she obtain in this way?
Two trapezoids are considered different if they are not congruent.
2010 Czech-Polish-Slovak Match, 3
Let $p$ be a prime number. Prove that from a $p^2\times p^2$ array of squares, we can select $p^3$ of the squares such that the centers of any four of the selected squares are not the vertices of a rectangle with sides parallel to the edges of the array.
2024 Singapore Senior Math Olympiad, Q4
Suppose $p$ is a prime number and $x, y, z$ are integers satisfying $0 < x < y < z <p$. If $x^3, y^3, z^3$ have equal remainders when divided by $p$, prove that $x ^ 2 + y ^ 2 + z ^ 2$ is divisible by $x + y + z$.
2014 Ukraine Team Selection Test, 8
The quadrilateral $ABCD$ is inscribed in the circle $\omega$ with the center $O$. Suppose that the angles $B$ and $C$ are obtuse and lines $AD$ and $BC$ are not parallel. Lines $AB$ and $CD$ intersect at point $E$. Let $P$ and $R$ be the feet of the perpendiculars from the point $E$ on the lines $BC$ and $AD$ respectively. $Q$ is the intersection point of $EP$ and $AD, S$ is the intersection point of $ER$ and $BC$. Let K be the midpoint of the segment $QS$ . Prove that the points $E, K$, and $O$ are collinear.
2013 South East Mathematical Olympiad, 8
$n\geq 3$ is a integer. $\alpha,\beta,\gamma \in (0,1)$. For every $a_k,b_k,c_k\geq0(k=1,2,\dotsc,n)$ with $\sum\limits_{k=1}^n(k+\alpha)a_k\leq \alpha, \sum\limits_{k=1}^n(k+\beta)b_k\leq \beta, \sum\limits_{k=1}^n(k+\gamma)c_k\leq \gamma$, we always have $\sum\limits_{k=1}^n(k+\lambda)a_kb_kc_k\leq \lambda$.
Find the minimum of $\lambda$
2009 India Regional Mathematical Olympiad, 5
A convex polygon is such that the distance between any two vertices does not exceed $ 1$.
$ (i)$ Prove that the distance between any two points on the boundary of the polygon does not exceed $ 1$.
$ (ii)$ If $ X$ and $ Y$ are two distinct points inside the polygon, prove that there exists a point $ Z$ on the boundary of the polygon such that $ XZ \plus{} YZ\le1$.
2022 Sharygin Geometry Olympiad, 9.8
Several circles are drawn on the plane and all points of their intersection or touching are marked. Is it possible that each circle contains exactly five marked points and each point belongs to exactly five circles?
2019 Kosovo National Mathematical Olympiad, 2
Show that when the product of three conscutive numbers we add arithmetic mean of them it is a perfect cube.
2007 AMC 12/AHSME, 23
How many non-congruent right triangles with positive integer leg lengths have areas that are numerically equal to $ 3$ times their perimeters?
$ \textbf{(A)}\ 6 \qquad \textbf{(B)}\ 7 \qquad \textbf{(C)}\ 8 \qquad \textbf{(D)}\ 10 \qquad \textbf{(E)}\ 12$
2014 Harvard-MIT Mathematics Tournament, 15
Given a regular pentagon of area $1$, a pivot line is a line not passing through any of the pentagon's vertices such that there are $3$ vertices of the pentagon on one side of the line and $2$ on the other. A pivot point is a point inside the pentagon with only finitely many non-pivot lines passing through it. Find the area of the region of pivot points.
Brazil L2 Finals (OBM) - geometry, 2020.5
Let $ABC$ be a triangle and $M$ the midpoint of $AB$. Let circumcircles of triangles $CMO$ and $ABC$ intersect at $K$ where $O$ is the circumcenter of $ABC$. Let $P$ be the intersection of lines $OM$ and $CK$. Prove that $\angle{PAK} = \angle{MCB}$.
2020/2021 Tournament of Towns, P1
Is it possible that a product of 9 consecutive positive integers is equal to a sum of 9 consecutive (not necessarily the same) positive integers?
[i]Boris Frenkin[/i]
2006 AMC 8, 19
Triangle $ ABC$ is an isosceles triangle with $ \overline{AB} \equal{}\overline{BC}$. Point $ D$ is the midpoint of both $ \overline{BC}$ and $ \overline{AE}$, and $ \overline{CE}$ is 11 units long. Triangle $ ABD$ is congruent to triangle $ ECD$. What is the length of $ \overline{BD}$?
[asy]size(100);
draw((0,0)--(2,4)--(4,0)--(6,4)--cycle--(4,0),linewidth(1));
label("$A$", (0,0), SW);
label("$B$", (2,4), N);
label("$C$", (4,0), SE);
label("$D$", shift(0.2,0.1)*intersectionpoint((0,0)--(6,4),(2,4)--(4,0)), N);
label("$E$", (6,4), NE);[/asy]
$ \textbf{(A)}\ 4 \qquad
\textbf{(B)}\ 4.5 \qquad
\textbf{(C)}\ 5 \qquad
\textbf{(D)}\ 5.5 \qquad
\textbf{(E)}\ 6$
1994 Romania TST for IMO, 3:
Let $a_1, a_2, . . ., a_n$ be a finite sequence of $0$ and $1$. Under any two consecutive terms of this sequence $0$ is written if the digits are equal and $1$ is written otherwise. This way a new sequence of length $n -1$ is obtained.
By repeating this procedure $n - 1$ times one obtains a triangular table of $0$ and $1$. Find the maximum possible number of ones that can appear on this table
2018 AIME Problems, 2
Let $a_0 = 2$, $a_1 = 5$, and $a_2 = 8$, and for $n>2$ define $a_n$ recursively to be the remainder when $4(a_{n-1} + a_{n-2} + a_{n-3})$ is divided by $11$. Find $a_{2018}\cdot a_{2020}\cdot a_{2022}$.
1988 IMO Shortlist, 11
The lock of a safe consists of 3 wheels, each of which may be set in 8 different ways positions. Due to a defect in the safe mechanism the door will open if any two of the three wheels are in the correct position. What is the smallest number of combinations which must be tried if one is to guarantee being able to open the safe (assuming the "right combination" is not known)?
2017 Online Math Open Problems, 5
Henry starts with a list of the first 1000 positive integers, and performs a series of steps on the list. At each step, he erases any nonpositive integers or any integers that have a repeated digit, and then decreases everything in the list by 1. How many steps does it take for Henry's list to be empty?
[i]Proposed by Michael Ren[/i]
2023 Austrian MO Beginners' Competition, 1
Let $x, y, z$ be nonzero real numbers with $$\frac{x + y}{z}=\frac{y + z}{x}=\frac{z + x}{y}.$$
Determine all possible values of $$\frac{(x + y)(y + z)(z + x)}{xyz}.$$
[i](Walther Janous)[/i]
1972 Poland - Second Round, 3
The coordinates of the triangle's vertices in the Cartesian system $XOY$ are integers. Prove that the diameter of the circle circumscribed by this triangle is not greater than the product of the lengths of the triangle's sides.
1993 China National Olympiad, 1
Given an odd $n$, prove that there exist $2n$ integers $a_1,a_2,\cdots ,a_n$; $b_1,b_2,\cdots ,b_n$, such that for any integer $k$ ($0<k<n$), the following $3n$ integers:
$a_i+a_{i+1}, a_i+b_i, b_i+b_{i+k}$ ($i=1,2,\cdots ,n; a_{n+1}=a_1, b_{n+j}=b_j, 0<j<n$) are of different remainders on division by $3n$.
2007 Sharygin Geometry Olympiad, 11
A boy and his father are standing on a seashore. If the boy stands on his tiptoes, his eyes are at a height of $1$ m above sea-level, and if he seats on father’s shoulders, they are at a height of $2$ m. What is the ratio of distances visible for him in two eases?
(Find the answer to $0,1$, assuming that the radius of Earth equals $6000$ km.)
2016 Estonia Team Selection Test, 7
On the sides $AB, BC$ and $CA$ of triangle $ABC$, points $L, M$ and $N$ are chosen, respectively, such that the lines $CL, AM$ and $BN$ intersect at a common point O inside the triangle and the quadrilaterals $ALON, BMOL$ and $CNOM$ have incircles. Prove that
$$\frac{1}{AL\cdot BM} +\frac{1}{BM\cdot CN} +\frac{1}{CN \cdot AL} =\frac{1}{AN\cdot BL} +\frac{1}{BL\cdot CM} +\frac{1}{CM\cdot AN} $$