Found problems: 85335
2001 Estonia National Olympiad, 1
John had to solve a math problem in the class. While cleaning the blackboard, he accidentally erased a part of his problem as well: the text that remained on board was $37 \cdot(72 + 3x) = 14**45$, where $*$ marks an erased digit. Show that John can still solve his problem, knowing that $x$ is an integer
2024 Brazil Team Selection Test, 1
Let $ABC$ be an acute triangle. Let $D$ be a point on side $AB$ and $E$ be a point on side $AC$ such that lines $BC$ and $DE$ are parallel. Let $X$ be an interior point of $BCED$. Suppose rays $DX$ and $EX$ meet side $BC$ at points $P$ and $Q$, respectively, such that both $P$ and $Q$ lie between $B$ and $C$. Suppose that the circumcircles of triangles $BQX$ and $CPX$ intersect at a point $Y \neq X$. Prove that the points $A, X$, and $Y$ are collinear.
2017 Junior Regional Olympiad - FBH, 3
In acute triangle $ABC$ holds $\angle BAC=80^{\circ}$, and altitudes $h_a$ and $h_b$ intersect in point $H$. if $\angle AHB = 126^{\circ}$, which side is the smallest, and which is the biggest in $ABC$
2024 Pan-African, 1
Find all positive intgers $a,b$ and $c$ such that $\frac{a+b}{a+c}=\frac{b+c}{b+a}$ and $ab+bc+ca$ is a prime number
1997 French Mathematical Olympiad, Problem 5
Given two distinct points $A,B$ in the plane, for each point $C$ not on the line $AB$, we denote by $G$ and $I$ the centroid and incenter of the triangle $ABC$, respectively.
(a) For $0<\alpha<\pi$, let $\Gamma$ be the set of points $C$ in the plane such that $\angle\left(\overrightarrow{CA},\overrightarrow{CB}\right)=\alpha+2k\pi$ as an oriented angle, where $k\in\mathbb Z$. If $C$ describes $\Gamma$, show that points $G$ and $I$ also descibre arcs of circles, and determine these circles.
(b) Suppose that in addition $\frac\pi3<\alpha<\pi$. For which positions of $C$ in $\Gamma$ is $GI$ minimal?
(c) Let $f(\alpha)$ denote the minimal $GI$ from the part (b). Give $f(\alpha)$ explicitly in terms of $a=AB$ and $\alpha$. Find the minimum value of $f(\alpha)$ for $\alpha\in\left(\frac\pi3,\pi\right)$.
2021 Korea National Olympiad, P5
A real number sequence $a_1, \cdots ,a_{2021}$ satisfies the below conditions.
$$a_1=1, a_2=2, a_{n+2}=\frac{2a_{n+1}^2}{a_n+a_{n+1}} (1\leq n \leq 2019)$$
Let the minimum of $a_1, \cdots ,a_{2021}$ be $m$, and the maximum of $a_1, \cdots ,a_{2021}$ be $M$.
Let a 2021 degree polynomial $$P(x):=(x-a_1)(x-a_2) \cdots (x-a_{2021})$$
$|P(x)|$ is maximum in $[m, M]$ when $x=\alpha$. Show that $1<\alpha <2$.
2011 Czech-Polish-Slovak Match, 2
In convex quadrilateral $ABCD$, let $M$ and $N$ denote the midpoints of sides $AD$ and $BC$, respectively. On sides $AB$ and $CD$ are points $K$ and $L$, respectively, such that $\angle MKA=\angle NLC$. Prove that if lines $BD$, $KM$, and $LN$ are concurrent, then \[ \angle KMN = \angle BDC\qquad\text{and}\qquad\angle LNM=\angle ABD.\]
2019 Israel National Olympiad, 3
Six congruent isosceles triangles have been put together as described in the picture below. Prove that points M, F, C lie on one line.
[img]https://i.imgur.com/1LU5Zmb.png[/img]
2017-IMOC, N3
Find all functions $f:\mathbb N\to\mathbb N_0$ such that for all $m,n\in\mathbb N$,
\begin{align*}
f(mn)&=f(m)f(n)\\
f(m+n)&=\min(f(m),f(n))\qquad\text{if }f(m)\ne f(n)\end{align*}
2009 JBMO TST - Macedonia, 4
In every $1\times1$ cell of a rectangle board a natural number is written. In one step it is allowed the numbers written in every cell of arbitrary chosen row, to be doubled, or the numbers written in the cells of the arbitrary chosen column to be decreased by 1. Will after final number of steps all the numbers on the board be $0$?
2017 Swedish Mathematical Competition, 1
Xenia and Yagve take turns in playing the following game: A coin is placed on the first box in a row of nine cells. At each turn the player may choose to move the coin forward one step, move the coin forward four steps, or move coin back two steps. For a move to be allowed, the coin must land on one of them of nine cells. The winner is one who gets to move the coin to the last ninth cell. Who wins, given that Xenia makes the first move, and both players play optimally?
2021 Grand Duchy of Lithuania, 1
Prove that for any polynomial $f(x)$ (with real coefficients) there exist polynomials $g(x)$ and $h(x)$ (with real coefficients) such that $f(x) = g(h(x)) - h(g(x))$.
2000 Korea Junior Math Olympiad, 1
For arbitrary natural number $a$, show that $\gcd(a^3+1, a^7+1)=a+1$.
2015 Singapore MO Open, 2
A boy lives in a small island in which there are three roads at every junction. He starts
from his home and walks along the roads. At each junction he would choose to turn
to the road on his right or left alternatively, i.e., his choices would be . . ., left, right,
left,... Prove that he will eventually return to his home.
2009 India Regional Mathematical Olympiad, 4
Find the sum of all 3-digit natural numbers which contain at least one odd digit and at least one even digit.
2018 Pan African, 5
Let $a$, $b$, $c$ and $d$ be non-zero pairwise different real numbers such that
$$
\frac{a}{b} + \frac{b}{c} + \frac{c}{d} + \frac{d}{a} = 4 \text{ and } ac = bd.
$$
Show that
$$
\frac{a}{c} + \frac{b}{d} + \frac{c}{a} + \frac{d}{b} \leq -12
$$
and that $-12$ is the maximum.
2023 Kyiv City MO Round 1, Problem 4
Let's call a pair of positive integers $\overline{a_1a_2\ldots a_k}$ and $\overline{b_1b_2\ldots b_k}$ $k$-similar if all digits $a_1, a_2, \ldots, a_k , b_1 , b_2, \ldots, b_k$ are distinct, and there exist distinct positive integers $m, n$, for which the following equality holds:
$$a_1^m + a_2^m + \ldots + a_k^m = b_1^n + b_2^n + \ldots + b_k^n$$
For which largest $k$ do there exist $k$-similar numbers?
[i]Proposed by Oleksiy Masalitin[/i]
2024 LMT Fall, 23
Define $\overline{a}$ of a positive integer $a$ to be the number $a$ with its digits reversed. For example, $\overline{31564} = 46513.$ Find the sum of all positive integers $n \leq 100$ such that $(\overline{n})^2=\overline{n^2}.$ (Note: For a number that ends with a zero, like 450, the reverse would exclude the zero, so $\overline{450}=54$).
2011 AMC 8, 11
The graph shows the number of minutes studied by both Asha (black bar) and Sasha (grey bar) in one week. On the average, how many more minutes per day did Sasha study than Asha?
[asy]
size(300);
real i;
defaultpen(linewidth(0.8));
draw((0,140)--origin--(220,0));
for(i=1;i<13;i=i+1) {
draw((0,10*i)--(220,10*i));
}
label("$0$",origin,W);
label("$20$",(0,20),W);
label("$40$",(0,40),W);
label("$60$",(0,60),W);
label("$80$",(0,80),W);
label("$100$",(0,100),W);
label("$120$",(0,120),W);
path MonD=(20,0)--(20,60)--(30,60)--(30,0)--cycle,MonL=(30,0)--(30,70)--(40,70)--(40,0)--cycle,TuesD=(60,0)--(60,90)--(70,90)--(70,0)--cycle,TuesL=(70,0)--(70,80)--(80,80)--(80,0)--cycle,WedD=(100,0)--(100,100)--(110,100)--(110,0)--cycle,WedL=(110,0)--(110,120)--(120,120)--(120,0)--cycle,ThurD=(140,0)--(140,80)--(150,80)--(150,0)--cycle,ThurL=(150,0)--(150,110)--(160,110)--(160,0)--cycle,FriD=(180,0)--(180,70)--(190,70)--(190,0)--cycle,FriL=(190,0)--(190,50)--(200,50)--(200,0)--cycle;
fill(MonD,grey);
fill(MonL,lightgrey);
fill(TuesD,grey);
fill(TuesL,lightgrey);
fill(WedD,grey);
fill(WedL,lightgrey);
fill(ThurD,grey);
fill(ThurL,lightgrey);
fill(FriD,grey);
fill(FriL,lightgrey);
draw(MonD^^MonL^^TuesD^^TuesL^^WedD^^WedL^^ThurD^^ThurL^^FriD^^FriL);
label("M",(30,-5),S);
label("Tu",(70,-5),S);
label("W",(110,-5),S);
label("Th",(150,-5),S);
label("F",(190,-5),S);
label("M",(-25,85),W);
label("I",(-27,75),W);
label("N",(-25,65),W);
label("U",(-25,55),W);
label("T",(-25,45),W);
label("E",(-25,35),W);
label("S",(-26,25),W);[/asy]
$ \textbf{(A)}\ 6\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 9\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 12 $
1993 Dutch Mathematical Olympiad, 4
Let $ C$ be a circle with center $ M$ in a plane $ V$, and $ P$ be a point not on the circle $ C$.
$ (a)$ If $ P$ is fixed, prove that $ AP^2\plus{}BP^2$ is a constant for every diameter $ AB$ of the circle $ C$.
$ (b)$ Let $ AB$ be a fixed diameter of $ C$ and $ P$ a point on a fixed sphere $ S$ not intersecting $ V$. Determine the points $ P$ on $ S$ that minimize $ AP^2\plus{}BP^2$.
1996 AIME Problems, 4
A wooden cube, whose edges are one centimeter long, rests on a horizontal surface. Illuminated by a point source of light that is $x$ centimeters directly above an upper vertex, the cube casts a shadow on the horizontal surface. The area of the shadow, which does not inclued the area beneath the cube is 48 square centimeters. Find the greatest integer that does not exceed $1000x.$
2012 Lusophon Mathematical Olympiad, 6
A quadrilateral $ABCD$ is inscribed in a circle of center $O$. It is known that the diagonals $AC$ and $BD$ are perpendicular. On each side we build semicircles, externally, as shown in the figure.
a) Show that the triangles $AOB$ and $COD$ have the equal areas.
b) If $AC=8$ cm and $BD= 6$ cm, determine the area of the shaded region.
2019 South East Mathematical Olympiad, 4
Let $X$ be a $5\times 5$ matrix with each entry be $0$ or $1$. Let $x_{i,j}$ be the $(i,j)$-th entry of $X$ ($i,j=1,2,\hdots,5$). Consider all the $24$ ordered sequence in the rows, columns and diagonals of $X$ in the following:
\begin{align*}
&(x_{i,1}, x_{i,2},\hdots,x_{i,5}),\ (x_{i,5},x_{i,4},\hdots,x_{i,1}),\ (i=1,2,\hdots,5) \\
&(x_{1,j}, x_{2,j},\hdots,x_{5,j}),\ (x_{5,j},x_{4,j},\hdots,x_{1,j}),\ (j=1,2,\hdots,5) \\
&(x_{1,1},x_{2,2},\hdots,x_{5,5}),\ (x_{5,5},x_{4,4},\hdots,x_{1,1}) \\
&(x_{1,5},x_{2,4},\hdots,x_{5,1}),\ (x_{5,1},x_{4,2},\hdots,x_{1,5})
\end{align*}
Suppose that all of the sequences are different. Find all the possible values of the sum of all entries in $X$.
2025 NCJMO, 5
Each element of set $\mathcal{S}$ is colored with multiple colors. A $\textit{rainbow}$ is a subset of $\mathcal{S}$ which has amongst its elements at least $1$ color from each element of $\mathcal{S}$. A $\textit{minimal rainbow}$ is a rainbow where removing any single element gives a non-rainbow.
Prove that the union of all minimal rainbows is $\mathcal{S}$.
[i]Grisham Paimagam[/i]
2021 BMT, 6
Three distinct integers are chosen uniformly at random from the set
$$\{2021, 2022, 2023, 2024, 2025, 2026, 2027, 2028, 2029, 2030\}.$$
Compute the probability that their arithmetic mean is an integer.