Found problems: 85335
2013 NZMOC Camp Selection Problems, 5
Consider functions $f$ from the whole numbers (non-negative integers) to the whole numbers that have the following properties:
$\bullet$ For all $x$ and $y$, $f(xy) = f(x)f(y)$,
$\bullet$ $f(30) = 1$, and
$\bullet$ for any $n$ whose last digit is $7$, $f(n) = 1$.
Obviously, the function whose value at $n$ is $ 1$ for all $n$ is one such function. Are there any others? If not, why not, and if so, what are they?
2007 Today's Calculation Of Integral, 214
Find the area of the region surrounded by the two curves $ y=\sqrt{x},\ \sqrt{x}+\sqrt{y}=1$ and the $ x$ axis.
2021 Saudi Arabia Training Tests, 2
Let $ABC$ be an acute, non isosceles triangle with the orthocenter $H$, circumcenter $O$ and $AD$ is the diameter of $(O)$. Suppose that the circle $(AHD)$ meets the lines $AB, AC$ at $F$, respectively. Denote $J, K$ as orthocenter and nine- point center of $AEF$. Prove that $HJ \parallel BC$ and $KO = KH$.
1966 IMO Shortlist, 53
Prove that in every convex hexagon of area $S$ one can draw a diagonal that cuts off a triangle of area not exceeding $\frac{1}{6}S.$
2002 Tournament Of Towns, 5
Does there exist a regular triangular prism that can be covered (without overlapping) by different equilateral triangles? (One is allowed to bend the triangles around the edges of the prism.)
2015 PAMO, Problem 2
A convex hexagon $ABCDEF$ is such that
$$AB=BC \quad CD=DE \quad EF=FA$$
and
$$\angle ABC=2\angle AEC \quad \angle CDE=2\angle CAE \quad \angle EFA=2\angle ACE$$
Show that $AD$, $CF$ and $EB$ are concurrent.
2019 Ramnicean Hope, 1
Calculate $ \lim_{n\to\infty }\sum_{t=1}^n\frac{1}{n+t+\sqrt{n^2+nt}} . $
[i]D.M. Bătinețu[/i] and [i]Neculai Stanciu[/i]
1965 All Russian Mathematical Olympiad, 060
There is a lighthouse on a small island. Its lamp enlights a segment of a sea to the distance $a$. The light is turning uniformly, and the end of the segment moves with the speed $v$. Prove that a ship, whose speed doesn't exceed $v/8$ cannot arrive to the island without being enlightened.
2013 AMC 10, 19
In base $10$, the number $2013$ ends in the digit $3$. In base $9$, on the other hand, the same number is written as $(2676)_9$ and ends in the digit $6$. For how many positive integers $b$ does the base-$b$ representation of $2013$ end in the digit $3$?
$\textbf{(A) }6\qquad
\textbf{(B) }9\qquad
\textbf{(C) }13\qquad
\textbf{(D) }16\qquad
\textbf{(E) }18\qquad$
2005 National Olympiad First Round, 8
How many natural number triples $(x,y,z)$ are there such that $xyz = 10^6$?
$
\textbf{(A)}\ 568
\qquad\textbf{(B)}\ 784
\qquad\textbf{(C)}\ 812
\qquad\textbf{(D)}\ 816
\qquad\textbf{(E)}\ 824
$
2025 Balkan MO, 2
In an acute-angled triangle \(ABC\), \(H\) be the orthocenter of it and \(D\) be any point on the side \(BC\). The points \(E, F\) are on the segments \(AB, AC\), respectively, such that the points \(A, B, D, F\) and \(A, C, D, E\) are cyclic. The segments \(BF\) and \(CE\) intersect at \(P.\) \(L\) is a point on \(HA\) such that \(LC\) is tangent to the circumcircle of triangle \(PBC\) at \(C.\) \(BH\) and \(CP\) intersect at \(X\). Prove that the points \(D, X, \) and \(L\) lie on the same line.
[i]Proposed by Theoklitos Parayiou, Cyprus [/i]
LMT Team Rounds 2021+, 9
Let $r_1, r_2, ..., r_{2021}$ be the not necessarily real and not necessarily distinct roots of $x^{2022} + 2021x = 2022$. Let $S_i = r_i^{2021}+2022r_i$ for all $1 \le i \le 2021$. Find $\left|\sum^{2021}_{i=1} S_i \right| = |S_1 +S_2 +...+S_{2021}|$.
2006 Germany Team Selection Test, 2
Four real numbers $ p$, $ q$, $ r$, $ s$ satisfy $ p+q+r+s = 9$ and $ p^{2}+q^{2}+r^{2}+s^{2}= 21$. Prove that there exists a permutation $ \left(a,b,c,d\right)$ of $ \left(p,q,r,s\right)$ such that $ ab-cd \geq 2$.
2019 India PRMO, 13
Each of the numbers $x_1, x_2, \ldots, x_{101}$ is $\pm 1$. What is the smallest positive value of $\sum_{1\leq i <
j \leq 101} x_i x_j$ ?
2024 Harvard-MIT Mathematics Tournament, 10
A polynomial $f(x)$ with integer coefficients is called $\textit{splitty}$ if and only if for every prime $p$, there exists polynomials $g_p, h_p$ with integer coefficients with degrees strictly smaller than the the degree of $f$, such that all coefficients of $f-g_ph_p$ are divisible by $p$. Compute the sum of all positive integers $n \leq 100$ such that $x^4+16x^2+n$ is $\textit{splitty}$.
2014 Poland - Second Round, 5.
Circles $o_1$ and $o_2$ tangent to some line at points $A$ and $B$, respectively, intersect at points $X$ and $Y$ ($X$ is closer to the line $AB$). Line $AX$ intersects $o_2$ at point $P\neq X$. Tangent to $o_2$ at point $P$ intersects line $AB$ at point $Q$. Prove that $\sphericalangle XYB = \sphericalangle BYQ$.
2009 Indonesia TST, 1
Ati has $ 7$ pots of flower, ordered in $ P_1,P_2,P_3,P_4,P_5,P_6,P_7$. She wants to rearrange the position of those pots to $ B_1,B_2,B_2,B_3,B_4,B_5,B_6,B_7$ such that for every positive integer $ n<7$, $ B_1,B_2,\dots,B_n$ is not the permutation of $ P_1,P_2,\dots,P_7$. In how many ways can Ati do this?
2022 CCA Math Bonanza, I10
Let $\overline{AB}$ be a line segment of length 2, $C_1$ be the circle with diameter $\overline{AB}$, $C_0$ be the circle with radius 2 externally tangent to $C_1$ at $A$, and $C_2$ be the circle with radius 3 externally tangent to $C_1$ at $B$. Duck $D_1$ is located at point $B$, Duck $D_2$ is located on $C_2$, 270 degrees counterclockwise from $B$, and Duck $D_0$ is located on $C_0$, 270 degrees counterclockwise from $A$. At the same time, the ducks all start running counterclockwise around their corresponding circles, with each duck taking the same amount of time to complete a full lap around its circle. When the 3 ducks are first collinear, the distance between $D_0$ and $D_2$ can be expressed as $p\sqrt{q}$. Find $p+q$.
[i]2022 CCA Math Bonanza Individual Round #10[/i]
2006 France Team Selection Test, 1
In a $2\times n$ array we have positive reals s.t. the sum of the numbers in each of the $n$ columns is $1$. Show that we can select a number in each column s.t. the sum of the selected numbers in each row is at most $\frac{n+1}4$.
2014 Greece JBMO TST, 4
Givan the set $S = \{1,2,3,....,n\}$. We want to partition the set $S$ into three subsets $A,B,C$ disjoint (to each other) with $A\cup B\cup C=S$ , such that the sums of their elements $S_{A} S_{B} S_{C}$ to be equal .Examine if this is possible when:
a) $n=2014$
b) $n=2015 $
c) $n=2018$
1998 French Mathematical Olympiad, Problem 2
Let $(u_n)$ be a sequence of real numbers which satisfies
$$u_{n+2}=|u_{n+1}|-u_n\qquad\text{for all }n\in\mathbb N.$$Prove that there exists a positive integer $p$ such that $u_n=u_{n+p}$ holds for all $n\in\mathbb N$.
2002 AMC 10, 16
Two walls and the ceiling of a room meet at right angles at point $P$. A fly is in the air one meter from one wall, eight meters from the other wall, and $9$ meters from point $P$. How many meters is the fly from the ceiling?
$\textbf{(A) }\sqrt{13}\qquad\textbf{(B) }\sqrt{14}\qquad\textbf{(C) }\sqrt{15}\qquad\textbf{(D) }4\qquad\textbf{(E) }\sqrt{17}$
2013 Princeton University Math Competition, 2
Find all pairs of positive integers $(a,b)$ such that:
\[\dfrac{a^3+4b}{a+2b^2+2a^2b}\]
is a positive integer.
2013 Canada National Olympiad, 3
Let $G$ be the centroid of a right-angled triangle $ABC$ with $\angle BCA = 90^\circ$. Let $P$ be the point on ray $AG$ such that $\angle CPA = \angle CAB$, and let $Q$ be the point on ray $BG$ such that $\angle CQB = \angle ABC$. Prove that the circumcircles of triangles $AQG$ and $BPG$ meet at a point on side $AB$.
2021 Junior Macedonian Mathematical Olympiad, Problem 1
At this year's Olympiad, some of the students are friends (friendship is symmetric), however there are also students which are not friends. No matter how the students are partitioned in two contest halls, there are always two friends in different halls. Let $A$ be a fixed student. Show that there exist students $B$ and $C$ such that there are exactly two friendships in the group $\{ A,B,C \}$.
[i]Authored by Mirko Petrushevski[/i]