Found problems: 85335
1999 India Regional Mathematical Olympiad, 1
Prove that the inradius of a right angled triangle with integer sides is an integer.
1989 Flanders Math Olympiad, 3
Show that:\[\alpha = \pm \frac{\pi}{12} + k\cdot \frac{\pi}2
(k\in \mathbb{Z}) \Longleftrightarrow\ |{\tan \alpha}| + |{\cot
\alpha}| = 4\]
2005 iTest, 5
The following is a code and is meant to be broken.
2 707 156 377 38 2 328 17 185 2 713 73 566 1130 328 73 38 259 471 38 17 566 2 134 707 38 274 377 328 38 1130 40 377 566 73 820 566 566 134 11 2 328 38 185 2 713 566 134 328 2 918 134 11 713 134 274 707 713 73 38 1130 17 134 707 11 820 707 707 38 17 713 73 38 134 566 40 2 918 377 566 134 713 38 328 820 274 4 38 566 707
156 377 38 707 40 2 918 377 566 134 713 38 328 820 274 4 38 566 134 707 713 73 38 2 328 707 991 38 566 713 377 713 73 38 707 38 918 38 328 713 73 707 73 377 566 713 2 328 707 991 38 566 532 820 38 707 713 134 377 328 377 328 713 73 134 707 713 38 707 713
185 2 713 73 566 1130 328 707 40 2 918 377 566 134 713 38 328 820 274 4 38 566 134 707 713 73 38 2 328 707 991 38 566 713 377 713 73 38 707 38 11 377 328 17 259 377 328 79 2 328 707 991 38 566 532 820 38 707 713 134 377 328 377 328 713 73 134 707 713 38 707 713
991 73 2 713 134 707 713 73 38 707 820 274 377 40 713 73 38 134 566 40 2 918 377 566 134 713 38 328 820 274 4 38 566 707
2012 Online Math Open Problems, 8
An $8 \times 8 \times 8$ cube is painted red on $3$ faces and blue on $3$ faces such that no corner is surrounded by three faces of the same color. The cube is then cut into $512$ unit cubes. How many of these cubes contain both red and blue paint on at least one of their faces?
[i]Author: Ray Li[/i]
[hide="Clarification"]The problem asks for the number of cubes that contain red paint on at least one face and blue paint on at least one other face, not for the number of cubes that have both colors of paint on at least one face (which can't even happen.)[/hide]
2016 IOM, 1
Find all positive integers $n$ such that there exist $n$ consecutive positive integers whose sum is a perfect square.
2008 Putnam, A3
Start with a finite sequence $ a_1,a_2,\dots,a_n$ of positive integers. If possible, choose two indices $ j < k$ such that $ a_j$ does not divide $ a_k$ and replace $ a_j$ and $ a_k$ by $ \gcd(a_j,a_k)$ and $ \text{lcm}\,(a_j,a_k),$ respectively. Prove that if this process is repeated, it must eventually stop and the final sequence does not depend on the choices made. (Note: $ \gcd$ means greatest common divisor and lcm means least common multiple.)
2018 Iran Team Selection Test, 3
$n>1$ and distinct positive integers $a_1,a_2,\ldots,a_{n+1}$ are given. Does there exist a polynomial $p(x)\in\Bbb{Z}[x]$ of degree $\le n$ that satisfies the following conditions?
a. $\forall_{1\le i < j\le n+1}: \gcd(p(a_i),p(a_j))>1 $
b. $\forall_{1\le i < j < k\le n+1}: \gcd(p(a_i),p(a_j),p(a_k))=1 $
[i]Proposed by Mojtaba Zare[/i]
2020 Princeton University Math Competition, A3/B5
Let $\{x\} = x- \lfloor x \rfloor$ . Consider a function f from the set $\{1, 2, . . . , 2020\}$ to the half-open interval $[0, 1)$. Suppose that for all $x, y,$ there exists a $z$ so that $\{f(x) + f(y)\} = f(z)$. We say that a pair of integers $m, n$ is valid if $1 \le m, n \le 2020$ and there exists a function $f$ satisfying the above so $f(1) = \frac{m}{n}$. Determine the sum over all valid pairs $m, n$ of ${m}{n}$.
2014 Lithuania Team Selection Test, 4
(a) Is there a natural number $n$ such that the number $2^n$ has last digit $6$ and the sum of the other digits is $2$?
b) Are there natural numbers $a$ and $m\ge 3$ such that the number $a^m$ has last digit $6$ and the sum of the other digits is 3?
2018 China Team Selection Test, 3
In isosceles $\triangle ABC$, $AB=AC$, points $D,E,F$ lie on segments $BC,AC,AB$ such that $DE\parallel AB$, $DF\parallel AC$. The circumcircle of $\triangle ABC$ $\omega_1$ and the circumcircle of $\triangle AEF$ $\omega_2$ intersect at $A,G$. Let $DE$ meet $\omega_2$ at $K\neq E$. Points $L,M$ lie on $\omega_1,\omega_2$ respectively such that $LG\perp KG, MG\perp CG$. Let $P,Q$ be the circumcenters of $\triangle DGL$ and $\triangle DGM$ respectively. Prove that $A,G,P,Q$ are concyclic.
2023 UMD Math Competition Part I, #17
The lengths of the sides of triangle $A'B'C'$ are equal to the lengths of the three medians of triangle $ABC.$ Then the ratio $\mathrm{Area} (A'B'C') / \mathrm{Area} (ABC)$ equals
$$
\mathrm a. ~ \frac 12\qquad \mathrm b.~\frac 23\qquad \mathrm c. ~\frac34 \qquad \mathrm d. ~\frac56 \qquad \mathrm e. ~\text{Cannot be determined from the information given.}
$$
2005 MOP Homework, 3
Suppose that $p$ and $q$ are distinct primes and $S$ is a subset of $\{1, 2, ..., p-1\}$. Let $N(S)$ denote the number of ordered $q$-tuples $(x_1,x_2,...,x_q)$ with $x_i \in S$, $1 \le i \le q$, such that $x_1+x_2+...+x_q \cong 0 (mod p)$.
2017 BMT Spring, 7
There are $86400$ seconds in a day, which can be deduced from the conversions between seconds, minutes, hours, and days. However, the leading scientists decide that we should decide on $3$ new integers $x, y$, and $z$, such that there are $x$ seconds in a minute, $y$ minutes in an hour, and $z$ hours in a day, such that $xyz = 86400$ as before, but such that the sum $x + y + z$ is minimized. What is the smallest possible value of that sum?
2007 Indonesia TST, 4
Let $ S$ be a finite family of squares on a plane such that every point on that plane is contained in at most $ k$ squares in $ S$. Prove that $ P$ can be divided into $ 4(k\minus{}1)\plus{}1$ sub-family such that in each sub-family, each pair of squares are disjoint.
2018 India IMO Training Camp, 3
Sir Alex plays the following game on a row of 9 cells. Initially, all cells are empty. In each move, Sir Alex is allowed to perform exactly one of the following two operations:
[list=1]
[*] Choose any number of the form $2^j$, where $j$ is a non-negative integer, and put it into an empty cell.
[*] Choose two (not necessarily adjacent) cells with the same number in them; denote that number by $2^j$. Replace the number in one of the cells with $2^{j+1}$ and erase the number in the other cell.
[/list]
At the end of the game, one cell contains $2^n$, where $n$ is a given positive integer, while the other cells are empty. Determine the maximum number of moves that Sir Alex could have made, in terms of $n$.
[i]Proposed by Warut Suksompong, Thailand[/i]
1992 Swedish Mathematical Competition, 5
A triangle has sides $a, b, c$ with longest side $c$, and circumradius $R$. Show that if $a^2 + b^2 = 2cR$, then the triangle is right-angled.
2012 Indonesia TST, 2
Let $P_1, P_2, \ldots, P_n$ be distinct $2$-element subsets of $\{1, 2, \ldots, n\}$. Suppose that for every $1 \le i < j \le n$, if $P_i \cap P_j \neq \emptyset$, then there is some $k$ such that $P_k = \{i, j\}$. Prove that if $a \in P_i$ for some $i$, then $a \in P_j$ for exactly one value of $j$ not equal to $i$.
2003 China National Olympiad, 3
Given a positive integer $n$, find the least $\lambda>0$ such that for any $x_1,\ldots x_n\in \left(0,\frac{\pi}{2}\right)$, the condition $\prod_{i=1}^{n}\tan x_i=2^{\frac{n}{2}}$ implies $\sum_{i=1}^{n}\cos x_i\le\lambda$.
[i]Huang Yumin[/i]
2018 Germany Team Selection Test, 1
Let $a_1,a_2,\ldots a_n,k$, and $M$ be positive integers such that
$$\frac{1}{a_1}+\frac{1}{a_2}+\cdots+\frac{1}{a_n}=k\quad\text{and}\quad a_1a_2\cdots a_n=M.$$
If $M>1$, prove that the polynomial
$$P(x)=M(x+1)^k-(x+a_1)(x+a_2)\cdots (x+a_n)$$
has no positive roots.
2010 VTRMC, Problem 7
Let $\sum_{n=1}^\infty a_n$ be a convergent series of positive terms (so $a_i>0$ for all $i$) and set $b_n=\frac1{na_n^2}$ for $n\ge1$. Prove that $\sum_{n=1}^\infty\frac n{b_1+b_2+\ldots+b_n}$ is convergent.
1966 IMO Longlists, 20
Given three congruent rectangles in the space. Their centers coincide, but the planes they lie in are mutually perpendicular. For any two of the three rectangles, the line of intersection of the planes of these two rectangles contains one midparallel of one rectangle and one midparallel of the other rectangle, and these two midparallels have different lengths. Consider the convex polyhedron whose vertices are the vertices of the rectangles.
[b]a.)[/b] What is the volume of this polyhedron ?
[b]b.)[/b] Can this polyhedron turn out to be a regular polyhedron ? If yes, what is the condition for this polyhedron to be regular ?
2017 Federal Competition For Advanced Students, P2, 4
(a) Determine the maximum $M$ of $x+y +z$ where $x, y$ and $z$ are positive real numbers with $16xyz = (x + y)^2(x + z)^2$.
(b) Prove the existence of infinitely many triples $(x, y, z)$ of positive rational numbers that satisfy $16xyz = (x + y)^2(x + z)^2$ and $x + y + z = M$.
Proposed by Karl Czakler
1954 Polish MO Finals, 2
What algebraic relationship holds between $ A $, $ B $, and $ C $ if
$$ctg A + \frac{\cos B}{\sin A \cos C} = ctg B + \frac{\cos A}{\sin B \cos C}.$$
Novosibirsk Oral Geo Oly VIII, 2023.6
Let's call a convex figure, the boundary of which consists of two segments and an arc of a circle, a mushroom-gon (see fig.). An arbitrary mushroom-gon is given. Use a compass and straightedge to draw a straight line dividing its area in half.
[img]https://cdn.artofproblemsolving.com/attachments/d/e/e541a83a7bb31ba14b3637f82e6a6d1ea51e22.png[/img]
1989 AMC 12/AHSME, 25
In a certain cross-country meet between two teams of five runners each, a runner who finishes in the $n^{th}$ position contributes $n$ to his team's score. The team with the lower score wins. If there are no ties among the runners, how many different winning scores are possible?
$ \textbf{(A)}\ 10 \qquad\textbf{(B)}\ 13 \qquad\textbf{(C)}\ 27 \qquad\textbf{(D)}\ 120 \qquad\textbf{(E)}\ 126 $