Found problems: 85335
2024 AIME, 4
Jen randomly picks $4$ distinct elements from $\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$. The lottery machine also picks $4$ distinct elements. If the lottery machine picks at least $2$ of Jen’s numbers, Jen wins a prize. If the lottery machine’s numbers are all $4$ of Jen’s, Jen wins the Grand Prize. Given that Jen wins a prize, what is the probability she wins a Grand Prize?
1996 Bulgaria National Olympiad, 1
Find all prime numbers $p,q$ for which $pq$ divides $(5^p-2^p)(5^q-2^q)$.
2007 Estonia Team Selection Test, 2
Let $D$ be the foot of the altitude of triangle $ABC$ drawn from vertex $A$. Let $E$ and $F$ be points symmetric to $D$ w.r.t. lines $AB$ and $AC$, respectively. Let $R_1$ and $R_2$ be the circumradii of triangles $BDE$ and $CDF$, respectively, and let $r_1$ and $r_2$ be the inradii of the same triangles. Prove that $|S_{ABD} - S_{ACD}| > |R_1r_1 - R_2r_2|$
2019 Brazil Undergrad MO, 3
Let $a,b,c$ be constants and $a,b,c$ are positive real numbers. Prove that the equations
$2x+y+z=\sqrt{c^2+z^2}+\sqrt{c^2+y^2}$
$x+2y+z=\sqrt{b^2+x^2}+\sqrt{b^2+z^2}$
$x+y+2z=\sqrt{a^2+x^2}+\sqrt{a^2+y^2}$
have exactly one real solution $(x,y,z)$ with $x,y,z \geq 0$.
2005 Harvard-MIT Mathematics Tournament, 2
How many real numbers $x$ are solutions to the following equation? \[ 2003^x + 2004^x = 2005^x \]
2018 Peru EGMO TST, 1
The number $n$ is "good", if there is three divisors of $n$($d_1, d_2, d_3$), such that $d_1^2+d_2^2+d_3^2=n$
a) Prove that all good number is divisible by $3$
b) Determine if there are infinite good numbers.
2006 Pre-Preparation Course Examination, 3
a) If $K$ is a finite extension of the field $F$ and $K=F(\alpha,\beta)$ show that $[K: F]\leq [F(\alpha): F][F(\beta): F]$
b) If $gcd([F(\alpha): F],[F(\beta): F])=1$ then does the above inequality always become equality?
c) By giving an example show that if $gcd([F(\alpha): F],[F(\beta): F])\neq 1$ then equality might happen.
2011 Morocco National Olympiad, 4
Let $ABC$ be a triangle with area $1$ and $P$ the middle of the side $[BC]$. $M$ and $N$ are two points of $[AB]-\left \{ A,B \right \} $ and $[AC]-\left \{ A,C \right \}$ respectively such that $AM=2MB$ and$CN=2AN$. The two lines $(AP)$ and $(MN)$ intersect in a point $D$. Find the area of the triangle $ADN$.
PEN K Problems, 5
Find all functions $f: \mathbb{N}\to \mathbb{N}$ such that for all $n\in \mathbb{N}$: \[f(f(m)+f(n))=m+n.\]
2023 Princeton University Math Competition, 11
11. Let $f(z)=\frac{a z+b}{c z+d}$ for $a, b, c, d \in \mathbb{C}$. Suppose that $f(1)=i, f(2)=i^{2}$, and $f(3)=i^{3}$. If the real part of $f(4)$ can be written as $\frac{m}{n}$ for relatively prime positive integers $m, n$, find $m^{2}+n^{2}$.
2015 IMO, 3
Let $ABC$ be an acute triangle with $AB > AC$. Let $\Gamma $ be its circumcircle, $H$ its orthocenter, and $F$ the foot of the altitude from $A$. Let $M$ be the midpoint of $BC$. Let $Q$ be the point on $\Gamma$ such that $\angle HQA = 90^{\circ}$ and let $K$ be the point on $\Gamma$ such that $\angle HKQ = 90^{\circ}$. Assume that the points $A$, $B$, $C$, $K$ and $Q$ are all different and lie on $\Gamma$ in this order.
Prove that the circumcircles of triangles $KQH$ and $FKM$ are tangent to each other.
Proposed by Ukraine
2024 ELMO Shortlist, G8
Let $ABC$ be a triangle, and let $D$ be a point on the internal angle bisector of $BAC$. Let $x$ be the ellipse with foci $B$ and $C$ passing through $D$, $y$ be the ellipse with foci $A$ and $C$ passing through $D$, and $z$ be the ellipse with foci $A$ and $B$ passing through $D$. Ellipses $x$ and $z$ intersect at distinct points $D$ and $E$, and ellipses $x$ and $y$ intersect at distinct points $D$ and $F$. Prove that $AD$ bisects angle $EAF$.
[i]Andrew Carratu[/i]
2007 AMC 10, 23
A pyramid with a square base is cut by a plane that is parallel to its base and is $ 2$ units from the base. The surface area of the smaller pyramid that is cut from the top is half the surface area of the original pyramid. What is the altitude of the original pyramid?
$ \textbf{(A)}\ 2\qquad
\textbf{(B)}\ 2 \plus{} \sqrt{2}\qquad
\textbf{(C)}\ 1 \plus{} 2\sqrt{2}\qquad
\textbf{(D)}\ 4\qquad
\textbf{(E)}\ 4 \plus{} 2\sqrt{2}$
2006 Turkey Team Selection Test, 3
If $x,y,z$ are positive real numbers and $xy+yz+zx=1$ prove that
\[ \frac{27}{4} (x+y)(y+z)(z+x) \geq ( \sqrt{x+y} +\sqrt{ y+z} + \sqrt{z+x} )^2 \geq 6 \sqrt 3. \]
2001 Turkey Team Selection Test, 2
Let $H$ be the intersection of the altitudes of an acute triangle $ABC$ and $D$ be the midpoint of $[AC]$. Show that $DH$ passes through one of the intersection point of the circumcircle of $ABC$ and the circle with diameter $[BH]$.
2005 Swedish Mathematical Competition, 3
In a triangle $ABC$ the bisectors of angles $A$ and $C$ meet the opposite sides at $D$ and $E$ respectively. Show that if the angle at $B$ is greater than $60^\circ$, then $AE +CD <AC$.
1955 AMC 12/AHSME, 24
The function $ 4x^2\minus{}12x\minus{}1$:
$ \textbf{(A)}\ \text{always increases as }x\text{ increases}\\
\textbf{(B)}\ \text{always decreases as }x\text{ decreases to 1} \\
\textbf{(C)}\ \text{cannot equal 0} \\
\textbf{(D)}\ \text{has a maximum value when }x\text{ is negative} \\
\textbf{(E)}\ \text{has a minimum value of \minus{}10}$
2020 South East Mathematical Olympiad, 7
Given any prime $p \ge 3$. Show that for all sufficient large positive integer $x$, at least one of $x+1,x+2,\cdots,x+\frac{p+3}{2}$ has a prime divisor greater than $p$.
1986 Vietnam National Olympiad, 1
Let $ ABCD$ be a square of side $ 2a$. An equilateral triangle $ AMB$ is constructed in the plane through $ AB$ perpendicular to the plane of the square. A point $ S$ moves on $ AB$ such that $ SB\equal{}x$. Let $ P$ be the projection of $ M$ on $ SC$ and $ E$, $ O$ be the midpoints of $ AB$ and $ CM$ respectively.
(a) Find the locus of $ P$ as $ S$ moves on $ AB$.
(b) Find the maximum and minimum lengths of $ SO$.
2013 Hong kong National Olympiad, 3
Let $ABC$ be a triangle with $CA>BC>AB$. Let $O$ and $H$ be the circumcentre and orthocentre of triangle $ABC$ respectively. Denote by $D$ and $E$ the midpoints of the arcs $AB$ and $AC$ of the circumcircle of triangle $ABC$ not containing the opposite vertices. Let $D'$ be the reflection of $D$ about $AB$ and $E'$ the reflection of $E$ about $AC$. Prove that $O,H,D',E'$ are concylic if and only if $A,D',E'$ are collinear.
2017 Novosibirsk Oral Olympiad in Geometry, 7
A car is driving along a straight highway at a speed of $60$ km per hour. Not far from the highway there is a parallel to him a $100$-meter fence. Every second, the passenger of the car measures the angle at which the fence is visible. Prove that the sum of all the angles he measured is less than $1100^o$
2004 239 Open Mathematical Olympiad, 5
The incircle of triangle $ABC$ touches its sides $AB, BC, CA$ in points $C_1, A_1, B_1$ respectively. The point $B_2$ is symmetric to $B_1$ with respect to line $A_1C_1$, lines $BB_2$ and $AC$ meet in point $B_3$. points $A_3$ and $C_3$ may be defined analogously. Prove that points $A_3, B_3$ and $C_3$ lie on a line, which passes through the circumcentre of a triangle $ABC$.
[b]
proposed by L. Emelyanov[/b]
1990 IMO Longlists, 15
Given a triangle $ ABC$. Let $ G$, $ I$, $ H$ be the centroid, the incenter and the orthocenter of triangle $ ABC$, respectively. Prove that $ \angle GIH > 90^{\circ}$.
1999 Korea Junior Math Olympiad, 1
There exists point $O$ inside a convex quadrilateral $ABCD$ satisfying $OA=OB$ and $OC=OD$, and $\angle AOB = \angle COD=90^{\circ}$. Consider two squares, (1)square having $AC$ as one side and located in the opposite side of $B$ and (2)square having $BD$ as one side and located in the opposite side of $E$. If the common part of these two squares is also a square, prove that $ABCD$ is an inscribed quadrilateral.
1986 Miklós Schweitzer, 6
Let $U$ denote the set $\{ f\in C[0, 1] \colon |f(x)|\leq 1\, \mathrm{for}\,\mathrm{all}\, x\in [0, 1]\}$. Prove that there is no topology on $C[0, 1]$ that, together with the linear structure of $C[0,1]$, makes $C[0,1]$ into a topological vector space in which the set $U$ is compact. (Assume that topological vector spaces are Hausdorff) [V. Totik]