Found problems: 85335
2000 Swedish Mathematical Competition, 4
The vertices of a triangle are three-dimensional lattice points. Show that its area is at least $\frac12$.
2016 India Regional Mathematical Olympiad, 2
Let $a,b,c$ be positive real numbers such that $$\frac{a}{1+a}+\frac{b}{1+b}+\frac{c}{1+c}=1.$$ Prove that $abc \le \frac{1}{8}$.
2010 Romania Team Selection Test, 5
Let $a$ and $n$ be two positive integer numbers such that the (positive) prime factors of $a$ be all greater than $n$.
Prove that $n!$ divides $(a - 1)(a^2 - 1)\cdots (a^{n-1} - 1)$.
[i]AMM Magazine[/i]
2024 Macedonian Balkan MO TST, Problem 4
Let $x_1, ..., x_n$ $(n \geq 2)$ be real numbers from the interval $[1,2]$. Prove that
$$|x_1-x_2|+...+|x_n-x_1| + \frac{1}{3} (|x_1-x_3|+...+|x_n-x_2|) \leq \frac{2}{3} (x_1+...+x_n)$$
and determine all cases of equality.
Kvant 2023, M2746
Turbo the snail sits on a point on a circle with circumference $1$. Given an infinite sequence of positive real numbers $c_1, c_2, c_3, \dots$, Turbo successively crawls distances $c_1, c_2, c_3, \dots$ around the circle, each time choosing to crawl either clockwise or counterclockwise.
Determine the largest constant $C > 0$ with the following property: for every sequence of positive real numbers $c_1, c_2, c_3, \dots$ with $c_i < C$ for all $i$, Turbo can (after studying the sequence) ensure that there is some point on the circle that it will never visit or crawl across.
2017 Taiwan TST Round 2, 2
Let $a,b,c,d$ be positive real numbers satisfying $a+b+c+d=4$. Prove that
$$\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{d}+\frac{d^2}{a}\geq 4+(a-d)^2$$
2024 IFYM, Sozopol, 6
A triangle \( ABC \) is given with centers \( O \) and \( I \) of the circumscribed and inscribed circles, respectively. Point \( A_1 \) is the reflection of \( A \) with respect to \( I \). Point \( A_2 \) is such that lines \( BA_1 \) and \( BA_2 \) are symmetric with respect to \( BI \), and lines \( CA_1 \) and \( CA_2 \) are symmetric with respect to \( CI \). Prove that \( AO^2 = |A_2O^2 - A_2I^2| \).
2022 239 Open Mathematical Olympiad, 5
Prove that there are infinitely many positive integers $k$ such that $k(k+1)(k+2)(k+3)$ has no prime divisor of the form $8t+5.$
2020 JBMO Shortlist, 5
The positive integer $k$ and the set $A$ of distinct integers from $1$ to $3k$ inclusively are such that there are no distinct $a$, $b$, $c$ in $A$ satisfying $2b = a + c$. The numbers from $A$ in the interval $[1, k]$ will be called [i]small[/i]; those in $[k + 1, 2k]$ - [i]medium[/i] and those in $[2k + 1, 3k]$ - [i]large[/i]. It is always true that there are [b]no[/b] positive integers $x$ and $d$ such that if $x$, $x + d$, and $x + 2d$ are divided by $3k$ then the remainders belong to $A$ and those of $x$ and $x + d$ are different and are:
a) small? $\hspace{1.5px}$ b) medium? $\hspace{1.5px}$ c) large?
([i]In this problem we assume that if a multiple of $3k$ is divided by $3k$ then the remainder is $3k$ rather than $0$[/i].)
2005 AMC 8, 3
What is the minimum number of small squares that must be colored black so that a line of symmetry lies on the diagonal $ \overline{BD}$ of square $ ABCD$?
[asy]defaultpen(linewidth(1));
for ( int x = 0; x < 5; ++x )
{
draw((0,x)--(4,x));
draw((x,0)--(x,4));
}
fill((1,0)--(2,0)--(2,1)--(1,1)--cycle);
fill((0,3)--(1,3)--(1,4)--(0,4)--cycle);
fill((2,3)--(4,3)--(4,4)--(2,4)--cycle);
fill((3,1)--(4,1)--(4,2)--(3,2)--cycle);
label("$A$", (0, 4), NW);
label("$B$", (4, 4), NE);
label("$C$", (4, 0), SE);
label("$D$", (0, 0), SW);[/asy]
$ \textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5 $
1962 Leningrad Math Olympiad, grade 8
[b]8.1[/b] Four circles are placed on planes so that each one touches the other two externally. Prove that the points of tangency lie on one circle.
[img]https://cdn.artofproblemsolving.com/attachments/9/8/883a82fb568954b09a4499a955372e2492dbb8.png[/img]
[b]8.2[/b]. Let the integers $a$ and $b$ be represented as $x^2-5y^2$, where $x$ and $y$ are integer numbers. Prove that the number $ab$ can also be presented in this form.
[b]8.3[/b] Solve the equation $x(x + d)(x + 2d)(x + 3d) = a$.
[b]8.4 / 9.1[/b] Let $a+b+c=1$, $m+n+p=1 $. Prove that $$-1 \le am + bn + cp \le 1 $$
[b]8.5[/b] Inscribe a triangle with the largest area in a semicircle.
[b]8.6[/b] Three circles of the same radius intersect at one point. Prove that the other three points intersections lie on a circle of the same radius.
[img]https://cdn.artofproblemsolving.com/attachments/4/7/014952f2dcf0349d54b07230e45a42c242a49d.png[/img]
[b]8.7[/b] Find the circle of smallest radius that contains a given triangle.
[b]8.8 / 9.2[/b] Given a polynomial $$x^{2n} +a_1x^{2n-2} + a_2x^{2n-4} + ... + a_{n-1}x^2 + a_n,$$ which is divisible by $ x-1$. Prove that it is divisible by $x^2-1$.
[b]8.9[/b] Prove that for any prime number $p$ other than $2$ and from $5$, there is a natural number $k$ such that only ones are involved in the decimal notation of the number $pk$..
PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3983459_1962_leningrad_math_olympiad]here[/url].
2022 Princeton University Math Competition, B1
Suppose that the greatest common divisor of $n$ and $5040$ is equal to $120.$ Determine the sum of the four smallest possible positive integers $n.$
2024 AIME, 5
Rectangles $ABCD$ and $EFGH$ are drawn such that $D,E,C,F$ are collinear. Also, $A,D,H,G$ all lie on a circle. If $BC=16,$ $AB=107,$ $FG=17,$ and $EF=184,$ what is the length of $CE$?
[asy]
unitsize(1 cm);
pair A, B, C, D, E, F, G, H;
A = (0,0);
B = (5,0);
C = (5,1.5);
D = (0,1.5);
E = (1,1.5);
F = (8,1.5);
G = (8,3.5);
H = (1,3.5);
draw(A--B--C--D--cycle);
draw(E--F--G--H--cycle);
dot("A", A, SW);
dot("B", B, SE);
dot("C", C, SE);
dot("D", D, NW);
dot("E", E, NW);
dot("F", F, SE);
dot("G", G, NE);
dot("H", H, NW);
[/asy]
2004 District Olympiad, 1
Let $n\geq 2$ and $1 \leq r \leq n$. Consider the set $S_r=(A \in M_n(\mathbb{Z}_2), rankA=r)$. Compute the sum $\sum_{X \in S_r}X$
BIMO 2022, 1
Given a graph $G$, consider the following two quantities,
$\bullet$ Assign to each vertex a number in $\{0,1,2\}$ such that for every edge $e=uv$, the numbers assigned to $u$ and $v$ have sum at least $2$. Let $A(G)$ be the minimum possible sum of the numbers written to each vertex satisfying this condition.
$\bullet$ Assign to each edge a number in $\{0,1,2\}$ such that for every vertex $v$, the sum of numbers on all edges containing $v$ is at most $2$. Let $B(G)$ be the maximum possible sum of the numbers written to each edge satisfying this condition.
Prove that $A(G)=B(G)$ for every graph $G$.
[Note: This question is not original]
[Extra: Show that this statement is still true if we replace $2$ to $n$, if and only if $n$ is even (where we replace $\{0,1,2\}$ to $\{0,1,\cdots, n\}$)]
2007 Estonia Team Selection Test, 6
Consider a $10 \times 10$ grid. On every move, we colour $4$ unit squares that lie in the intersection of some two rows and two columns. A move is allowed if at least one of the $4$ squares is previously uncoloured. What is the largest possible number of moves that can be taken to colour the whole grid?
2017 ELMO Shortlist, 2
An integer $n>2$ is called [i]tasty[/i] if for every ordered pair of positive integers $(a,b)$ with $a+b=n,$ at least one of $\frac{a}{b}$ and $\frac{b}{a}$ is a terminating decimal. Do there exist infinitely many tasty integers?
[i]Proposed by Vincent Huang[/i]
2023 Kyiv City MO, Problem 1
The rectangle is cut into 6 squares, as shown on the figure below. The gray square in the middle has a side equal to 1.
What is the area of the rectangle?
[img]https://i.ibb.co/gg1tBTN/Kyiv-MO-2023-7-1.png[/img]
2013 National Chemistry Olympiad, 48
In which set are both elements metalloids?
${ \textbf{(A)}\ \text{Cr and Mo}\qquad\textbf{(B)}\ \text{Ge and As}\qquad\textbf{(C)}\ \text{Sn and Pb}\qquad\textbf{(D)}}\ \text{Se and Br}\qquad $
1985 AMC 12/AHSME, 23
If \[x \equal{} \frac { \minus{} 1 \plus{} i\sqrt3}{2}\qquad\text{and}\qquad y \equal{} \frac { \minus{} 1 \minus{} i\sqrt3}{2},\] where $ i^2 \equal{} \minus{} 1$, then which of the following is [i]not[/i] correct?
$ \textbf{(A)}\ x^5 \plus{} y^5 \equal{} \minus{} 1 \qquad \textbf{(B)}\ x^7 \plus{} y^7 \equal{} \minus{} 1 \qquad \textbf{(C)}\ x^9 \plus{} y^9 \equal{} \minus{} 1$
$ \textbf{(D)}\ x^{11} \plus{} y^{11} \equal{} \minus{} 1 \qquad \textbf{(E)}\ x^{13} \plus{} y^{13} \equal{} \minus{} 1$
2013 Stanford Mathematics Tournament, 12
What is the greatest possible value of c such that $x^2+5x+c=0$ has at least one real solution?
2003 Purple Comet Problems, 13
Let $P(x)$ be a polynomial such that, when divided by $x - 2$, the remainder is $3$ and, when divided by $x - 3$, the remainder is $2$. If, when divided by $(x - 2)(x - 3)$, the remainder is $ax + b$, find $a^2 + b^2$.
1988 IMO Longlists, 57
$ S$ is the set of all sequences $ \{a_i| 1 \leq i \leq 7, a_i \equal{} 0 \text{ or } 1\}.$ The distance between two elements $ \{a_i\}$ and $ \{b_i\}$ of $ S$ is defined as
\[ \sum^7_{i \equal{} 1} |a_i \minus{} b_i|.
\]
$ T$ is a subset of $ S$ in which any two elements have a distance apart greater than or equal to 3. Prove that $ T$ contains at most 16 elements. Give an example of such a subset with 16 elements.
2018 EGMO, 1
Let $ABC$ be a triangle with $CA=CB$ and $\angle{ACB}=120^\circ$, and let $M$ be the midpoint of $AB$. Let $P$ be a variable point of the circumcircle of $ABC$, and let $Q$ be the point on the segment $CP$ such that $QP = 2QC$. It is given that the line through $P$ and perpendicular to $AB$ intersects the line $MQ$ at a unique point $N$.
Prove that there exists a fixed circle such that $N$ lies on this circle for all possible positions of $P$.
2021 Honduras National Mathematical Olympiad, Problem 3
Let $a$ and $b$ be positive integers satisfying
\[ \frac a{a-2} = \frac{b+2021}{b+2008} \]
Find the maximum value $\dfrac ab$ can attain.