Found problems: 85335
1988 National High School Mathematics League, 4
Given three planes $\alpha,\beta,\gamma$. Intersection angle between any two planes are all $\theta$.$\alpha\cap\beta=a,\beta\cap\gamma=b,\gamma\cap\alpha=c$.
Given two conditions:
A: $\theta>\frac{\pi}{3}$
B: $a,b,c$ share one point.
$(\text{A})$A is sufficient but unnecessary condition of B.
$(\text{B})$A is necessary but insufficient condition of B.
$(\text{C})$A is sufficient and necessary condition of B.
$(\text{D})$None above
2022 CMIMC Integration Bee, 13
\[\int_{-\infty}^\infty e^{-x^2-4/x^2}\,\mathrm dx\]
[i]Proposed by Vlad Oleksenko[/i]
2022 Taiwan TST Round 1, N
Show that $n!=a^{n-1}+b^{n-1}+c^{n-1}$ has only finitely many solutions in positive integers.
[i]Proposed by Dorlir Ahmeti, Albania[/i]
2012 China Northern MO, 3
Suppose $S= \{x|x=a^2+ab+b^2,a,b \in Z\}$. Prove that:
(1) If $m \in S$, $3|m$ , then $\frac{m}{3} \in S$
(2) If $m,n \in S$ , then $mn\in S$.
2013 Iran MO (3rd Round), 4
A polygon $A$ that doesn't intersect itself and has perimeter $p$ is called [b]Rotund[/b] if for each two points $x,y$ on the sides of this polygon which their distance on the plane is less than $1$ their distance on the polygon is at most $\frac{p}{4}$. (Distance on the polygon is the length of smaller path between two points on the polygon)
Now we shall prove that we can fit a circle with radius $\frac{1}{4}$ in any rotund polygon.
The mathematicians of two planets earth and Tarator have two different approaches to prove the statement. In both approaches by "inner chord" we mean a segment with both endpoints on the polygon, and "diagonal" is an inner chord with vertices of the polygon as the endpoints.
[b]Earth approach: Maximal Chord[/b]
We know the fact that for every polygon, there exists an inner chord $xy$ with a length of at most 1 such that for any inner chord $x'y'$ with length of at most 1 the distance on the polygon of $x,y$ is more than the distance on the polygon of $x',y'$. This chord is called the [b]maximal chord[/b].
On the rotund polygon $A_0$ there's two different situations for maximal chord:
(a) Prove that if the length of the maximal chord is exactly $1$, then a semicircle with diameter maximal chord fits completely inside $A_0$, so we can fit a circle with radius $\frac{1}{4}$ in $A_0$.
(b) Prove that if the length of the maximal chord is less than one we still can fit a circle with radius $\frac{1}{4}$ in $A_0$.
[b]Tarator approach: Triangulation[/b]
Statement 1: For any polygon that the length of all sides is less than one and no circle with radius $\frac{1}{4}$ fits completely inside it, there exists a triangulation of it using diagonals such that no diagonal with length more than $1$ appears in the triangulation.
Statement 2: For any polygon that no circle with radius $\frac{1}{4}$ fits completely inside it, can be divided into triangles that their sides are inner chords with length of at most 1.
The mathematicians of planet Tarator proved that if the second statement is true, for each rotund polygon there exists a circle with radius $\frac{1}{4}$ that fits completely inside it.
(c) Prove that if the second statement is true, then for each rotund polygon there exists a circle with radius $\frac{1}{4}$ that fits completely inside it.
They found out that if the first statement is true then the second statement is also true, so they put a bounty of a doogh on proving the first statement. A young earth mathematician named J.N., found a counterexample for statement 1, thus receiving the bounty.
(d) Find a 1392-gon that is counterexample for statement 1.
But the Tarators are not disappointed and they are still trying to prove the second statement.
(e) (Extra points) Prove or disprove the second statement.
Time allowed for this problem was 150 minutes.
2019 IberoAmerican, 4
Let $ABCD$ be a trapezoid with $AB\parallel CD$ and inscribed in a circumference $\Gamma$. Let $P$ and $Q$ be two points on segment $AB$ ($A$, $P$, $Q$, $B$ appear in that order and are distinct) such that $AP=QB$. Let $E$ and $F$ be the second intersection points of lines $CP$ and $CQ$ with $\Gamma$, respectively. Lines $AB$ and $EF$ intersect at $G$. Prove that line $DG$ is tangent to $\Gamma$.
2009 Tournament Of Towns, 2
Several points on the plane are given, no three of them lie on the same line. Some of these points are connected by line segments. Assume that any line that does not pass through any of these points intersects an even number of these segments. Prove that from each point exits an even number of the segments.
2017 Middle European Mathematical Olympiad, 2
Determine the smallest possible real constant $C$ such that the inequality
$$|x^3 + y^3 + z^3 + 1| \leq C|x^5 + y^5 + z^5 + 1|$$
holds for all real numbers $x, y, z$ satisfying $x + y + z = -1$.
2019 Tournament Of Towns, 5
Basil has an unrestricted supply of straight bricks $1 \times 1 \times 3$ and Γ-shape bricks made of three cubes $1\times 1\times 1$. Basil filled a whole box $m \times n \times k$ with these bricks, where $m, n$ and $k$ are integers greater than $1$. Prove that it was sufficient to use only Γ-shape bricks.
(Mikhail Evdokimov)
2014 Singapore Senior Math Olympiad, 24
Find the number of integers $x$ which satisfy the equation $(x^2-5x+5)^{x+5}=1$.
2023 Indonesia TST, N
Let $P(x)$ and $Q(x)$ be polynomials of degree $p$ and $q$ respectively such that every coefficient is $1$ or $2023$. If $P(x)$ divides $Q(x)$, prove that $p+1$ divides $q+1$.
2010 Germany Team Selection Test, 3
Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that
\[f(x)f(y) = (x+y+1)^2 \cdot f \left( \frac{xy-1}{x+y+1} \right)\] $\forall x,y \in \mathbb{R}$ with $x+y+1 \neq 0$ and $f(x) > 1$ $\forall x > 0.$
2022 BMT, 8
Define the two sequences $a_0, a_1, a_2, \cdots$ and $b_0, b_1, b_2, \cdots$ by $a_0 = 3$ and $b_0 = 1$ with the recurrence relations $a_{n+1} = 3a_n + b_n$ and $b_{n+1} = 3b_n - a_n$ for all nonnegative integers $n.$ Let $r$ and $s$ be the remainders when $a_{32}$ and $b_{32}$ are divided by $31,$ respectively. Compute $100r + s.$
1998 Croatia National Olympiad, Problem 1
Solve the equation $2z^3-(5+6i)z^2+9iz+1-3i=0$ over $\mathbb C$ given that one of the solutions is real.
1984 Czech And Slovak Olympiad IIIA, 4
Let $r$ be a natural number greater than $1$. Then there exist positive irrational numbers $x, y$ such that $x^y = r$ . Prove it.
2002 Iran MO (3rd Round), 25
An ant walks on the interior surface of a cube, he moves on a straight line. If ant reaches to an edge the he moves on a straight line on cube's net. Also if he reaches to a vertex he will return his path.
a) Prove that for each beginning point ant can has infinitely many choices for his direction that its path becomes periodic.
b) Prove that if if the ant starts from point $A$ and its path is periodic, then for each point $B$ if ant starts with this direction, then his path becomes periodic.
2017 Lusophon Mathematical Olympiad, 6
Let ABC be a scalene triangle. Consider points D, E, F on segments AB, BC, CA, respectively, such that $\overline{AF}$=$\overline{DF}$ and $\overline{BE}$=$\overline{DE}$.
Show that the circumcenter of ABC lies on the circumcircle of CEF.
1989 Vietnam National Olympiad, 1
Are there integers $ x$, $ y$, not both divisible by $ 5$, such that $ x^2 \plus{} 19y^2 \equal{} 198\cdot 10^{1989}$?
2015 Junior Balkan Team Selection Test, 1
Frog is in the origin of decartes coordinate system. Every second frog jumpes horizontally or vertically in some of the $4$ adjacent points which coordinates are integers. Find number of different points in which frog can be found in $2015$ seconds.
2013 Stanford Mathematics Tournament, 9
Big candles cost 16 cents and burn for exactly 16 minutes. Small candles cost 7 cents and burn for exactly 7 minutes. The candles burn at possibly varying and unknown rates, so it is impossible to predictably modify the amount of time for which a candle will burn except by burning it down for a known amount of time. Candles may be arbitrarily and instantly put out and relit. Compute the cost in cents of the cheapest set of big and small candles you need to measure exactly 1 minute.
2011 Bosnia And Herzegovina - Regional Olympiad, 3
Let $I$ be the incircle and $O$ a circumcenter of triangle $ABC$ such that $\angle ACB=30^{\circ}$. On sides $AC$ and $BC$ there are points $E$ and $D$, respectively, such that $EA=AB=BD$. Prove that $DE=IO$ and $DE \perp IO$
2016 Korea Junior Math Olympiad, 6
circle $O_1$ is tangent to $AC$, $BC$(side of triangle $ABC$) at point $D, E$.
circle $O_2$ include $O_1$, is tangent to $BC$, $AB$(side of triangle $ABC$) at point $E, F$
The tangent of $O_2$ at $P(DE \cap O_2, P \neq E)$ meets $AB$ at $Q$.
A line passing through $O_1$(center of $O_1$) and parallel to $BO_2$($O_2$ is also center of $O_2$) meets $BC$ at $G$, $EQ \cap AC=K, KG \cap EF=L$, $EO_2$ meets circle $O_2$ at $N(\neq E)$, $LO_2 \cap FN=M$.
IF $N$ is a middle point of $FM$, prove that $BG=2EG$
2011 Brazil Team Selection Test, 1
Let $ABC$ be an acute triangle with $D, E, F$ the feet of the altitudes lying on $BC, CA, AB$ respectively. One of the intersection points of the line $EF$ and the circumcircle is $P.$ The lines $BP$ and $DF$ meet at point $Q.$ Prove that $AP = AQ.$
[i]Proposed by Christopher Bradley, United Kingdom[/i]
2001 AIME Problems, 15
The numbers 1, 2, 3, 4, 5, 6, 7, and 8 are randomly written on the faces of a regular octahedron so that each face contains a different number. The probability that no two consecutive numbers, where 8 and 1 are considered to be consecutive, are written on faces that share an edge is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$
Durer Math Competition CD Finals - geometry, 2009.D1
Fencing Ferdinand wants to fence three rectangular areas. there are fences in three types, with $4$ amount of fences of each type. You will notice that there is always at least as much area it manages to enclose a total of three by enclosing three square areas (i.e., each area fencing elements of the same size to enclose it) as if it were three different, rectangular would encircle an area (i.e., use two different elements for each of the three areas). Why is this is so? When does it not matter how he fences the rectangles, in terms of the sum of the areas?