Found problems: 85335
2008 Hanoi Open Mathematics Competitions, 1
How many integers from $1$ to $2008$ have the sum of their digits divisible by $5$ ?
2019 Dutch IMO TST, 3
Find all functions $f : Z \to Z$ satisfying the following two conditions:
(i) for all integers $x$ we have $f(f(x)) = x$,
(ii) for all integers $x$ and y such that $x + y$ is odd, we have $f(x) + f(y) \ge x + y$.
2002 Stanford Mathematics Tournament, 3
A clockmaker wants to design a clock such that the area swept by each hand (second, minute, and hour) in one minute is the same (all hands move continuously). What is the length of the hour hand divided by the length of the second hand?
2015 239 Open Mathematical Olympiad, 1
There are 10 stones of different weights with distinct pairwise sums. We have a special two-tiered balance scale such that only two stones can be put on each cup and then we understand which cup is heavier. Prove that having this scale you can either find the heaviest or the lightest stone.
2015 AMC 10, 9
The shaded region below is called a shark's fin falcata, a figure studied by Leonardo da Vinci. It is bounded by the portion of the circle of radius $3$ and center $(0,0)$ that lies in the first quadrant, the portion of the circle with radius $\tfrac{3}{2}$ and center $(0,\tfrac{3}{2})$ that lies in the first quadrant, and the line segment from $(0,0)$ to $(3,0)$. What is the area of the shark's fin falcata?
[asy]
import cse5;pathpen=black;pointpen=black;
size(1.5inch);
D(MP("x",(3.5,0),S)--(0,0)--MP("\frac{3}{2}",(0,3/2),W)--MP("y",(0,3.5),W));
path P=(0,0)--MP("3",(3,0),S)..(3*dir(45))..MP("3",(0,3),W)--(0,3)..(3/2,3/2)..cycle;
draw(P,linewidth(2));
fill(P,gray);
[/asy]
$\textbf{(A) } \dfrac{4\pi}{5}
\qquad\textbf{(B) } \dfrac{9\pi}{8}
\qquad\textbf{(C) } \dfrac{4\pi}{3}
\qquad\textbf{(D) } \dfrac{7\pi}{5}
\qquad\textbf{(E) } \dfrac{3\pi}{2}
$
1980 AMC 12/AHSME, 6
A positive number $x$ satisfies the inequality $\sqrt{x} < 2x$ if and only if
$\text{(A)} \ x > \frac{1}{4} \qquad \text{(B)} \ x > 2 \qquad \text{(C)} x > 4 \qquad \text{(D)} \ x < \frac{1}{4}\qquad \text{(E)} x < 4$
2024 Serbia JBMO TST, 4
Let $I$ be the incenter of a triangle $ABC$ with $AB \neq AC$. Let $M$ be the midpoint of $BC$, $M' \in BC$ be such that $IM'=IM$ and $K$ be the midpoint of the arc $BAC$. If $AK \cap BC=L$, show that $KLIM'$ is cyclic.
2012 Online Math Open Problems, 35
Let $s(n)$ be the number of 1's in the binary representation of $n$. Find the number of ordered pairs of integers $(a,b)$ with $0 \leq a < 64, 0 \leq b < 64$ and $s(a+b) = s(a) + s(b) - 1$.
[i]Author:Anderson Wang[/i]
LMT Team Rounds 2021+, B19
Kevin is at the point $(19,12)$. He wants to walk to a point on the ellipse $9x^2 + 25y^2 = 8100$, and then walk to $(-24, 0)$. Find the shortest length that he has to walk.
[i]Proposed by Kevin Zhao[/i]
2003 District Olympiad, 3
(a) If $\displaystyle ABC$ is a triangle and $\displaystyle M$ is a point from its plane, then prove that
\[ \displaystyle AM \sin A \leq BM \sin B + CM \sin C . \]
(b) Let $\displaystyle A_1,B_1,C_1$ be points on the sides $\displaystyle (BC),(CA),(AB)$ of the triangle $\displaystyle ABC$, such that the angles of $\triangle A_1 B_1 C_1$ are $\widehat{A_1} = \alpha, \widehat{B_1} = \beta, \widehat{C_1} = \gamma$. Prove that
\[ \displaystyle \sum A A_1 \sin \alpha \leq \sum BC \sin \alpha . \]
[i]Dan Ştefan Marinescu, Viorel Cornea[/i]
2009 Purple Comet Problems, 7
The figure $ABCD$ is bounded by a semicircle $CDA$ and a quarter circle $ABC$. Given that the distance from $A$ to $C$ is $18$, find the area of the figure.
[asy]
size(200);
defaultpen(linewidth(0.8));
pair A=(-9,0),B=(0,9*sqrt(2)-9),C=(9,0),D=(0,9);
dot(A^^B^^C^^D);
draw(arc(origin,9,0,180)^^arc((0,-9),9*sqrt(2),45,135));
label("$A$",A,S);
label("$B$",B,N);
label("$C$",C,S);
label("$D$",D,N);
[/asy]
2008 ITest, 49
Wendy takes Honors Biology at school, a smallish class with only fourteen students (including Wendy) who sit around a circular table. Wendy's friends Lucy, Starling, and Erin are also in that class. Last Monday none of the fourteen students were absent from class. Before the teacher arrived, Lucy and Starling stretched out a blue piece of yarn between them. Then Wendy and Erin stretched out a red piece of yarn between them at about the same height so that the yarn would intersect if possible. If all possible positions of the students around the table are equally likely, let $m/n$ be the probability that the yarns intersect, where $m$ and $n$ are relatively prime positive integers. Compute $m+n$.
2007 IMC, 4
Let $ G$ be a finite group. For arbitrary sets $ U, V, W \subset G$, denote by $ N_{UVW}$ the number of triples $ (x, y, z) \in U \times V \times W$ for which $ xyz$ is the unity .
Suppose that $ G$ is partitioned into three sets $ A, B$ and $ C$ (i.e. sets $ A, B, C$ are pairwise disjoint and $ G = A \cup B \cup C$). Prove that $ N_{ABC}= N_{CBA}.$
1991 Arnold's Trivium, 89
Calculate the sum of vector products $[[x, y], z] + [[y, z], x] + [[z, x], y]$
2000 Harvard-MIT Mathematics Tournament, 46
For what integer values of $n$ is $1+n+\frac{n^2}{2}+\cdots +\frac{n^n}{n!}$ an integer?
2011 Albania National Olympiad, 2
Find all the values that can take the last digit of a "perfect" even number. (The natural number $n$ is called "perfect" if the sum of all its natural divisors is equal twice the number itself.For example: the number $6$ is perfect ,because $1+2+3+6=2\cdot6$).
2013 National Olympiad First Round, 11
How many pairs of real numbers $(x,y)$ are there such that $x^4+y^4 + 2x^2y + 2xy^2+ 2 = x^2 + y^2 + 2x + 2y$?
$
\textbf{(A)}\ 6
\qquad\textbf{(B)}\ 5
\qquad\textbf{(C)}\ 4
\qquad\textbf{(D)}\ 3
\qquad\textbf{(E)}\ 2
$
India EGMO 2025 TST, 2
Two positive integers are called anagrams if every decimal digit occurs the same number of times in each of them (not counting the leading zeroes). Find all non-constant polynomials $P$ with non-negative integer coefficients so that whenever $a$ and $b$ are anagrams, $P(a)$ and $P(b)$ are anagrams as well.
Proposed by Sutanay Bhattacharya
2022 Saudi Arabia BMO + EGMO TST, 2.2
Given is an acute triangle $ABC$ with $BC < CA < AB$. Points $K$ and $L$ lie on segments $AC$ and $AB$ and satisfy $AK = AL = BC$. Perpendicular bisectors of segments $CK$ and $BL$ intersect line $BC$ at points $P$ and $Q$, respectively. Segments $KP$ and $LQ$ intersect at $M$. Prove that $CK + KM = BL + LM$.
2021 AMC 12/AHSME Spring, 24
Semicircle $\Gamma$ has diameter $\overline{AB}$ of length $14$. Circle $\Omega$ lies tangent to $\overline{AB}$ at a point $P$ and intersects $\Gamma$ at points $Q$ and $R$. If $QR=3\sqrt3$ and $\angle QPR=60^\circ$, then the area of $\triangle PQR$ is $\frac{a\sqrt{b}}{c}$, where $a$ and $c$ are relatively prime positive integers, and $b$ is a positive integer not divisible by the square of any prime. What is $a+b+c$?
$\textbf{(A) }110 \qquad \textbf{(B) }114 \qquad \textbf{(C) }118 \qquad \textbf{(D) }122\qquad \textbf{(E) }126$
2020 ISI Entrance Examination, 7
Consider a right-angled triangle with integer-valued sides $a<b<c$ where $a,b,c$ are pairwise co-prime. Let $d=c-b$ . Suppose $d$ divides $a$ . Then
[b](a)[/b] Prove that $d\leqslant 2$.
[b](b)[/b] Find all such triangles (i.e. all possible triplets $a,b,c$) with perimeter less than $100$ .
2018 Dutch IMO TST, 3
Let $n \ge 0$ be an integer. A sequence $a_0,a_1,a_2,...$ of integers is de fined as follows:
we have $a_0 = n$ and for $k \ge 1, a_k$ is the smallest integer greater than $a_{k-1}$ for which $a_k +a_{k-1}$ is the square of an integer.
Prove that there are exactly $\lfloor \sqrt{2n}\rfloor$ positive integers that cannot be written in the form $a_k - a_{\ell}$ with $k > \ell\ge 0$.
2003 Vietnam National Olympiad, 1
Find the largest positive integer $n$ such that the following equations have integer solutions in $x, y_{1}, y_{2}, ... , y_{n}$ :
$(x+1)^{2}+y_{1}^{2}= (x+2)^{2}+y_{2}^{2}= ... = (x+n)^{2}+y_{n}^{2}.$
2020 USMCA, 24
Farmer John has a $47 \times 53$ rectangular square grid. He labels the first row $1, 2, \cdots, 47$, the second row $48, 49, \cdots, 94$, and so on. He plants corn on any square of the form $47x + 53y$, for non-negative integers $x, y$. Given that the unplanted squares form a contiguous region $R$, find the perimeter of $R$.
2021 AIME Problems, 6
For any finite set $S$, let $|S|$ denote the number of elements in $S$. FInd the number of ordered pairs $(A,B)$ such that $A$ and $B$ are (not necessarily distinct) subsets of $\{1,2,3,4,5\}$ that satisfy
$$|A| \cdot |B| = |A \cap B| \cdot |A \cup B|$$