Found problems: 85335
2025 CMIMC Team, 2
We are searching for the number $7$ in the following binary tree:
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We use the following algorithm (which terminates with probability $1$):
[list=1]
[*] Write down the number currently at the root node.
[*] If we wrote down $7,$ terminate.
[*] Else, pick a random edge, and swap the two numbers at the endpoints of that edge
[*] Go back to step $1.$
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Let $p(a)$ be the probability that we ever write down the number $a$ after running the algorithm once. Find $$p(1)+p(2)+p(3)+p(5)+p(6).$$
Kvant 2023, M2743
Perimeter of triangle $ABC$ is $1$. Circle $\omega$ touches side $BC$, continuation of side $AB$ at $P$ and continuation of side $AC$ in $Q$. Line through midpoints $AB$ and $AC$ intersects circumcircle of $APQ$ at $X$ and $Y$.
Find length of $XY$.
2008 National Olympiad First Round, 28
A unit square from one of the corners of a $8\times 8$ chessboard is cut and thrown. At least how many triangles are necessary to divide the new board into triangles with equal areas?
$
\textbf{(A)}\ 17
\qquad\textbf{(B)}\ 19
\qquad\textbf{(C)}\ 20
\qquad\textbf{(D)}\ 21
\qquad\textbf{(E)}\ \text{None of the above}
$
2017 239 Open Mathematical Olympiad, 4
A polynomial $f(x)$ with integer coefficients is given. We define $d(a,k)=|f^k(a)-a|.$ It is known that for each integer $a$ and natural number $k$, $d(a,k)$ is positive. Prove that for all such $a,k$, $$d(a,k) \geq \frac{k}{3}.$$ ($f^k(x)=f(f^{k-1}(x)), f^0(x)=x.$)
2012 Bundeswettbewerb Mathematik, 1
given a positive integer $n$.
the set $\{ 1,2,..,2n \}$ is partitioned into $a_1<a_2<...<a_n $ and $b_1>b_2>...>b_n$.
find the value of : $ \sum_{i=1}^{n}|a_i - b_i| $
2018 AMC 12/AHSME, 4
A circle has a chord of length $10$, and the distance from the center of the circle to the chord is $5$. What is the area of the circle?
$\textbf{(A) }25\pi\qquad\textbf{(B) }50\pi\qquad\textbf{(C) }75\pi\qquad\textbf{(D) }100\pi\qquad\textbf{(E) }125\pi$
Kvant 2021, M2653
Let $p{}$ and $q{}$ be two coprime positive integers. A frog hops along the integer line so that on every hop it moves either $p{}$ units to the right or $q{}$ units to the left. Eventually, the frog returns to the initial point. Prove that for every positive integer $d{}$ with $d < p + q$ there are two numbers visited by the frog which differ just by $d{}$.
[i]Nikolay Belukhov[/i]
Russian TST 2018, P4
Let $k$ be a given even positive integer. Sarah first picks a positive integer $N$ greater than $1$ and proceeds to alter it as follows: every minute, she chooses a prime divisor $p$ of the current value of $N$, and multiplies the current $N$ by $p^k -p^{-1}$ to produce the next value of $N$. Prove that there are infinitely many even positive integers $k$ such that, no matter what choices Sarah makes, her number $N$ will at some point be divisible by $2018$.
2006 China Team Selection Test, 1
Let $k$ be an odd number that is greater than or equal to $3$. Prove that there exists a $k^{th}$-degree integer-valued polynomial with non-integer-coefficients that has the following properties:
(1) $f(0)=0$ and $f(1)=1$; and.
(2) There exist infinitely many positive integers $n$ so that if the following equation: \[ n= f(x_1)+\cdots+f(x_s), \] has integer solutions $x_1, x_2, \dots, x_s$, then $s \geq 2^k-1$.
2008 AMC 10, 15
How many right triangles have integer leg lengths $ a$ and $ b$ and a hypotenuse of length $ b\plus{}1$, where $ b<100$?
$ \textbf{(A)}\ 6 \qquad
\textbf{(B)}\ 7 \qquad
\textbf{(C)}\ 8 \qquad
\textbf{(D)}\ 9 \qquad
\textbf{(E)}\ 10$
2019 India Regional Mathematical Olympiad, 2
Given a circle $\tau$, let $P$ be a point in its interior, and let $l$ be a line through $P$. Construct with proof using ruler and compass, all circles which pass through $P$, are tangent to $\tau$ and whose center lies on line $l$.
2022 Math Prize for Girls Problems, 19
Let $S_-$ be the semicircular arc defined by
\[
(x + 1)^2 + (y - \frac{3}{2})^2 = \frac{1}{4} \text{ and } x \le -1.
\]
Let $S_+$ be the semicircular arc defined by
\[
(x - 1)^2 + (y - \frac{3}{2})^2 = \frac{1}{4} \text{ and } x \ge 1.
\]
Let $R$ be the locus of points $P$ such that $P$ is the intersection of two lines, one of the form $Ax + By = 1$ where $(A, B) \in S_-$ and the other of the form $Cx + Dy = 1$ where $(C, D) \in S_+$. What is the area of $R$?
2011 Tournament of Towns, 3
In a convex quadrilateral $ABCD, AB = 10, BC = 14, CD = 11$ and $DA = 5$. Determine the angle between its diagonals.
2011 IFYM, Sozopol, 2
On side $AB$ of $\Delta ABC$ is chosen point $M$. A circle is tangent internally to the circumcircle of $\Delta ABC$ and segments $MB$ and $MC$ in points $P$ and $Q$ respectively. Prove that the center of the inscribed circle of $\Delta ABC$ lies on line $PQ$.
2019 Azerbaijan Senior NMO, 4
Is it possible to construct a equilateral triangle such that:
$\text{a)}$ Coordinates of this triangle are integers in two dimensional plane?
$\text{b)}$ Coordinates of this triangle are integers in three dimensional plane?
2008 ITest, 46
Let $S$ be the sum of all $x$ in the interval $[0,2\pi)$ that satisfy \[\tan^2 x - 2\tan x\sin x=0.\] Compute $\lfloor10S\rfloor$.
2016 Putnam, A6
Find the smallest constant $C$ such that for every real polynomial $P(x)$ of degree $3$ that has a root in the interval $[0,1],$
\[\int_0^1|P(x)|\,dx\le C\max_{x\in[0,1]}|P(x)|.\]
2009 Princeton University Math Competition, 3
It is known that a certain mechanical balance can measure any object of integer mass anywhere between 1 and 2009 (both included). This balance has $k$ weights of integral values. What is the minimum $k$ for which there exist weights that satisfy this condition?
2021 Sharygin Geometry Olympiad, 19
A point $P$ lies inside a convex quadrilateral $ABCD$. Common internal tangents to the incircles of triangles $PAB$ and $PCD$ meet at point $Q$, and common internal tangents to the incircles of $PBC,PAD$ meet at point $R$. Prove that $P,Q,R$ are collinear.
1991 IberoAmerican, 2
A square is divided in four parts by two perpendicular lines, in such a way that three of these parts have areas equal to 1. Show that the square has area equal to 4.
2016 China Team Selection Test, 3
Let $n \geq 2$ be a natural. Define
$$X = \{ (a_1,a_2,\cdots,a_n) | a_k \in \{0,1,2,\cdots,k\}, k = 1,2,\cdots,n \}$$.
For any two elements $s = (s_1,s_2,\cdots,s_n) \in X, t = (t_1,t_2,\cdots,t_n) \in X$, define
$$s \vee t = (\max \{s_1,t_1\},\max \{s_2,t_2\}, \cdots , \max \{s_n,t_n\} )$$
$$s \wedge t = (\min \{s_1,t_1 \}, \min \{s_2,t_2,\}, \cdots, \min \{s_n,t_n\})$$
Find the largest possible size of a proper subset $A$ of $X$ such that for any $s,t \in A$, one has $s \vee t \in A, s \wedge t \in A$.
2023 Malaysian IMO Training Camp, 4
Find the largest constant $c>0$ such that for every positive integer $n\ge 2$, there always exist a positive divisor $d$ of $n$ such that $$d\le \sqrt{n}\hspace{0.5cm} \text{and} \hspace{0.5cm} \tau(d)\ge c\sqrt{\tau(n)}$$ where $\tau(n)$ is the number of divisors of $n$.
[i]Proposed by Mohd. Suhaimi Ramly[/i]
2018 EGMO, 4
A domino is a $ 1 \times 2 $ or $ 2 \times 1 $ tile.
Let $n \ge 3 $ be an integer. Dominoes are placed on an $n \times n$ board in such a way that each domino covers exactly two cells of the board, and dominoes do not overlap. The value of a row or column is the number of dominoes that cover at least one cell of this row or column. The configuration is called balanced if there exists some $k \ge 1 $ such that each row and each column has a value of $k$. Prove that a balanced configuration exists for every $n \ge 3 $, and find the minimum number of dominoes needed in such a configuration.
2004 AMC 10, 4
A standard six-sided die is rolled, and $ P$ is the product of the fi ve numbers that are visible. What is the largest number that is certain to divide $ P$?
$ \textbf{(A)}\ 6\qquad
\textbf{(B)}\ 12\qquad
\textbf{(C)}\ 24\qquad
\textbf{(D)}\ 144\qquad
\textbf{(E)}\ 720$
1971 AMC 12/AHSME, 15
An aquarium on a level table has rectangular faces and is $10$ inches wide and $8$ inches high. When it was tilted, the water in it covered an $8"\times 10"$ end but only three-fourths of the rectangular room. The depth of the water when the bottom was again made level, was
$\textbf{(A) }2\textstyle{\frac{1}{2}}"\qquad\textbf{(B) }3"\qquad\textbf{(C) }3\textstyle{\frac{1}{4}}"\qquad\textbf{(D) }3\textstyle{\frac{1}{2}}"\qquad \textbf{(E) }4"$