Found problems: 85335
2008 ITest, 19
Let $A$ be the set of positive integers that are the product of two consecutive integers. Let $B$ the set of positive integers that are the product of three consecutive integers. Find the sum of the two smallest elements of $A\cap B$.
2021 JHMT HS, 9
Squares of side lengths $1,$ $2,$ $3,$ and $4,$ are placed on a line segment $\ell$ from left to right, respectively, and these squares lie on the same side of $\ell,$ forming a polygon $P.$ An equilateral triangle whose base is $\ell$ is drawn around the squares such that its other two sides intersect $P$ at its leftmost and rightmost vertices (that are not on $\ell$). The area of the triangle can be written in the form $\tfrac{a + b\sqrt{3}}{c},$ where $a,$ $b,$ and $c$ are positive integers, and $b$ and $c$ are relatively prime. Find $a + b + c.$
2021 Malaysia IMONST 1, 18
How many real numbers $x$ are solutions to the equation $|x - 2| - 4 =\frac{1}{|x - 3|}$ ?
2012 Princeton University Math Competition, B3
Find, with proof, all pairs $(x, y)$ of integers satisfying the equation $3x^2+ 4 = 2y^3$.
2020 SJMO, 6
We say a positive integer $n$ is [i]$k$-tasty[/i] for some positive integer $k$ if there exists a permutation $(a_0, a_1, a_2, \ldots , a_n)$ of $(0,1,2, \ldots, n)$ such that $|a_{i+1} - a_i| \in \{k, k+1\}$ for all $0 \le i \le n-1$. Prove that for all positive integers $k$, there exists a constant $N$ such that all integers $n \geq N$ are $k$-tasty.
[i]Proposed by Anthony Wang[/i]
2003 Korea Junior Math Olympiad, 5
Four odd positive intgers $a, b, c, d (a\leq b \leq c\leq d)$ are given. Choose any three numbers among them and divide their sum by the un-chosen number, and you will always get the remainder as $1$. Find all $(a, b, c, d)$ that satisfies this.
2019 CCA Math Bonanza, L1.3
Points $P$ and $Q$ are chosen on diagonal $AC$ of square $ABCD$ such that $AB=AP=CQ=1$. What is the measure of $\angle{PBQ}$ in degrees?
[i]2019 CCA Math Bonanza Lightning Round #1.3[/i]
2016 Postal Coaching, 2
Find all $n \in \mathbb N$ such that $n = \varphi (n) + 402$, where $\varphi$ denotes the Euler phi function.
Russian TST 2015, P1
Prove that there exist two natural numbers $a,b$ such that $|a-m|+|b-n|>1000$ for any relatively prime natural numbers $m,n$.
2008 Flanders Math Olympiad, 3
A quadrilateral pyramid and a regular tetrahedron have edges that are all equal in length. They are glued together so that they have in common $1$ equilateral triangle . Prove that the resulting body has exactly $5$ sides.
2025 Bulgarian Spring Mathematical Competition, 9.3
In a country, there are towns, some of which are connected by roads. There is a route (not necessarily direct) between every two towns. The Minister of Education has ensured that every town without a school is connected via a direct road to a town that has a school. The Minister of State Optimization wants to ensure that there is a unique path between any two towns (without repeating traveled segments), which may require removing some roads.
Is it always possible to achieve this without constructing additional schools while preserving what the Minister of Education has accomplished?
MIPT Undergraduate Contest 2019, 1.3
Given a natural number $n$, for what maximal value $k$ it is possible to construct a matrix of size $k \times n$ consisting only of elements $\pm 1$ in such a way that for any interchange of a $+1$ with a $-1$ or vice versa, its rank is equal to $k$?
1994 Romania TST for IMO, 2:
Let $S_1, S_2,S_3$ be spheres of radii $a, b, c$ respectively whose centers lie on a line $l$. Sphere $S_2$ is externally tangent to $S_1$ and $S_3$, whereas $S_1$ and $S_3$ have no common points. A straight line t touches each of the spheres,
Find the sine of the angle between $l$ and $t$
2016 Kyiv Mathematical Festival, P5
On the board a 20-digit number which have 10 ones and 10 twos in its decimal form is written. It is allowed to choose two different digits and to reverse the order of digits in the interval between them. Is it always possible to get a number divisible by 11 using such operations?
2011 District Olympiad, 4
Find all the functions $f:[0,1]\rightarrow \mathbb{R}$ for which we have:
\[|x-y|^2\le |f(x)-f(y)|\le |x-y|,\]
for all $x,y\in [0,1]$.
2017 German National Olympiad, 4
Let $ABCD$ be a cyclic quadrilateral. The point $P$ is chosen on the line $AB$ such that the circle passing through $C,D$ and $P$ touches the line $AB$. Similarly, the point $Q$ is chosen on the line $CD$ such that the circle passing through $A,B$ and $Q$ touches the line $CD$.
Prove that the distance between $P$ and the line $CD$ equals the distance between $Q$ and $AB$.
2005 Morocco National Olympiad, 4
$21$ distinct numbers are chosen from the set $\{1,2,3,\ldots,2046\}.$ Prove that we can choose three distinct numbers $a,b,c$ among those $21$ numbers such that
\[bc<2a^2<4bc\]
2008 Philippine MO, 3
Let $P$ be a point outside a circle $\Gamma$, and let the two tangent lines through $P$ touch $\Gamma$ at $A$ and $B$. Let $C$ be on the minor arc $AB$, and let ray $PC$ intersect $\Gamma$ again at $D$. Let $\ell$ be the line through $B$ and parallel to $PA$. $\ell$ intersects $AC$ and $AD$ at $E$ and $F$, respectively. Prove that $B$ is the midpoint of $EF$.
2014 Purple Comet Problems, 1
The diagram below shows a circle with center $F$. The angles are related with $\angle BFC = 2\angle AFB$, $\angle CFD = 3\angle AFB$, $\angle DFE = 4\angle AFB$, and $\angle EFA = 5\angle AFB$. Find the degree measure of $\angle BFC$.
[asy]
size(4cm);
pen dps = fontsize(10);
defaultpen(dps);
dotfactor=4;
draw(unitcircle);
pair A,B,C,D,E,F;
A=dir(90);
B=dir(66);
C=dir(18);
D=dir(282);
E=dir(210);
F=origin;
dot("$F$",F,NW);
dot("$A$",A,dir(90));
dot("$B$",B,dir(66));
dot("$C$",C,dir(18));
dot("$D$",D,dir(306));
dot("$E$",E,dir(210));
draw(F--E^^F--D^^F--C^^F--B^^F--A);
[/asy]
2020 European Mathematical Cup, 2
Let $n$ and $k$ be positive integers. An $n$-tuple $(a_1, a_2,\ldots , a_n)$ is called a permutation if every number from the set $\{1, 2, . . . , n\}$ occurs in it exactly once. For a permutation $(p_1, p_2, . . . , p_n)$, we define its $k$-mutation to be the $n$-tuple
$$(p_1 + p_{1+k}, p_2 + p_{2+k}, . . . , p_n + p_{n+k}),$$
where indices are taken modulo $n$. Find all pairs $(n, k)$ such that every two distinct permutations have distinct $k$-mutations.
[i]Remark[/i]: For example, when $(n, k) = (4, 2)$, the $2$-mutation of $(1, 2, 4, 3)$ is $(1 + 4, 2 + 3, 4 + 1, 3 + 2) = (5, 5, 5, 5)$.
[i]Proposed by Borna Šimić[/i]
2022 Belarus - Iran Friendly Competition, 1
Do there exist a sequence $a_1, a_2, \ldots , a_n, \ldots$ of positive integers such that for any
positive integers $i, j$:
$$d(a_i + a_j ) = i + j?$$
Here $d(n)$ is the number of positive divisors of a positive integer
1992 Irish Math Olympiad, 1
Describe in geometric terms the set of points $(x,y)$ in the plane such that $x$ and $y$ satisfy the condition $t^2+yt+x\ge 0$ for all $t$ with $-1\le t\le 1$.
Denmark (Mohr) - geometry, 1998.1
In the figure shown, the small circles have radius $1$. Calculate the area of the gray part of the figure.
[img]https://1.bp.blogspot.com/-oy-WirJ6u9o/XzcFc3roVDI/AAAAAAAAMX8/qxNy5I_0RWUOxl-ZE52fnrwo0v0T7If9QCLcBGAsYHQ/s0/1998%2BMohr%2Bp1.png[/img]
2012 Lusophon Mathematical Olympiad, 4
An ant decides to walk on the perimeter of an $ABC$ triangle. The ant can start at any vertex. Whenever the ant is in a vertex, it chooses one of the adjacent vertices and walks directly (in a straight line) to the chosen vertex.
a) In how many ways can the ant walk around each vertex exactly twice?
b) In how many ways can the ant walk around each vertex exactly three times?
Note: For each item, consider that the starting vertex is visited.
2022 HMNT, 24
A string consisting of letters $A, C, G,$ and $U$ is [i]untranslatable[/i] if and only if it has no $\text{AUG}$ as a consecutive substring. For example, $\text{ACUGG}$ is untranslatable.
Let $a_n$ denote the number of untranslatable strings of length $n.$ It is given that there exists a unique triple of real numbers $(x,y,z)$ such that $a_n = xa_{n-1} + ya_{n-2} +za_{n-3}$ for all integers $n \ge 100.$ Compute $(x, y,z)$