Found problems: 85335
2015 Gulf Math Olympiad, 2
a) Let $UVW$ , $U'V'W'$ be two triangles such that $ VW = V'W' , UV = U'V' , \angle WUV = \angle W'U'V'.$
Prove that the angles $\angle VWU , \angle V'W'U'$ are equal or supplementary.
b) $ABC$ is a triangle where $\angle A$ is [b]obtuse[/b]. take a point $P$ inside the triangle , and extend $AP,BP,CP$ to meet the sides $BC,CA,AB$ in $K,L,M$ respectively. Suppose that $PL = PM .$
1) If $AP$ bisects $\angle A$ , then prove that $AB = AC$ .
2) Find the angles of the triangle $ABC$ if you know that $AK,BL,CM$ are angle bisectors of the triangle $ABC$ and that $2AK = BL$.
2003 Kurschak Competition, 1
Draw a circle $k$ with diameter $\overline{EF}$, and let its tangent in $E$ be $e$. Consider all possible pairs $A,B\in e$ for which $E\in \overline{AB}$ and $AE\cdot EB$ is a fixed constant. Define $(A_1,B_1)=(AF\cap k,BF\cap k)$. Prove that the segments $\overline{A_1B_1}$ all concur in one point.
2007 Indonesia TST, 4
Given a collection of sets $X = \{A_1, A_2, ..., A_n\}$. A set $\{a_1, a_2, ..., a_n\}$ is called a single representation of $X$ if $a_i \in A_i$ for all i. Let $|S| = mn$, $S = A_1\cup A_2 \cup ... \cup A_n = B_1 \cup B_2 \cup ... \cup B_n$ with $|A_i| = |B_i| = m$ for all $i$. Prove that $S = C_1 \cup C_2 \cup ... \cup C_n$ where for every $i, C_i $ is a single represenation for $\{A_j\}_{j=1}^n $and $\{B_j\}_{j=1}^n$.
2024 Thailand TST, 3
Elisa has $2023$ treasure chests, all of which are unlocked and empty at first. Each day, Elisa adds a new gem to one of the unlocked chests of her choice, and afterwards, a fairy acts according to the following rules:
[list=disc]
[*]if more than one chests are unlocked, it locks one of them, or
[*]if there is only one unlocked chest, it unlocks all the chests.
[/list]
Given that this process goes on forever, prove that there is a constant $C$ with the following property: Elisa can ensure that the difference between the numbers of gems in any two chests never exceeds $C$, regardless of how the fairy chooses the chests to unlock.
2007 Flanders Math Olympiad, 3
Let $ABCD$ be a square with side $10$. Let $M$ and $N$ be the midpoints of $[AB]$ and $[BC]$ respectively. Three circles are drawn: one with midpoint $D$ and radius $|AD|$, one with midpoint $M$ and radius $|AM|$, and one with midpoint $N$ and radius $|BN|$. The three circles intersect in the points $R, S$ and $T$ inside the square. Determine the area of $\triangle RST$.
2018 Switzerland - Final Round, 5
Does there exist any function $f: \mathbb{R}^+ \to \mathbb{R}$ such that for every positive real number $x,y$ the following is true :
$$f(xf(x)+yf(y)) = xy$$
2024 India Regional Mathematical Olympiad, 3
Let $ABC$ be an acute triangle with $AB = AC$. Let $D$ be the point on $BC$ such that $AD$ is perpendicular to $BC$. Let $O,H,G$ be the circumcenter, orthocenter and centroid of triangle $ABC$ respectively. Suppose that $2 \cdot OD = 23 \cdot HD$. Prove that $G$ lies on the incircle of triangle $ABC$.
2016 Saudi Arabia IMO TST, 3
Let $n \ge 4$ be a positive integer and there exist $n$ positive integers that are arranged on a circle such that:
$\bullet$ The product of each pair of two non-adjacent numbers is divisible by $2015 \cdot 2016$.
$\bullet$ The product of each pair of two adjacent numbers is not divisible by $2015 \cdot 2016$.
Find the maximum value of $n$
2024 Azerbaijan BMO TST, 6
Let $ABC$ be an acute triangle ($AB < BC < AC$) with circumcircle $\Gamma$. Assume there exists $X \in AC$ satisfying $AB=BX$ and $AX=BC$. Points $D, E \in \Gamma$ are taken such that $\angle ADB<90^{\circ}$, $DA=DB$ and $BC=CE$. Let $P$ be the intersection point of $AE$ with the tangent line to $\Gamma$ at $B$, and let $Q$ be the intersection point of $AB$ with tangent line to $\Gamma$ at $C$. Show that the projection of $D$ onto $PQ$ lies on the circumcircle of $\triangle PAB$.
VI Soros Olympiad 1999 - 2000 (Russia), 11.5
Find all polynomials $P(x)$ with real coefficients such that for all real $x$ holds the equality $$(1 + 2x)P(2x) = (1 + 2^{1999}x)P(x) .$$
1961 Putnam, A3
Evaluate
$$\lim_{n\to \infty} \sum_{j=1}^{n^{2}} \frac{n}{n^2 +j^2 }.$$
2013 AMC 12/AHSME, 1
On a particular January day, the high temperature in Lincoln, Nebraska, was 16 degrees higher than the low temperature, and the average of the high and low temperatures was $3^{\circ}$. In degrees, what was the low temperature in Lincoln that day?
$\textbf{(A) }-13\qquad\textbf{(B) }-8\qquad\textbf{(C) }-5\qquad\textbf{(D) }3\qquad\textbf{(E) }11$
1922 Eotvos Mathematical Competition, 3
Show that, if $a,b,...,n$ are distinct natural numbers, none divisible by any primes greater than $3$, then
$$\frac{1}{a}+\frac{1}{b}+...+ \frac{1}{n}< 3$$
1998 Romania National Olympiad, 2
Show that there is no positive integer $n$ such that $n + k^2$ is a perfect square for at least $n$ positive integer values of $k$.
1975 AMC 12/AHSME, 2
For which real values of $ m$ are the simultaneous equations
\begin{align*} y &= mx + 3 \\
y &= (2m - 1)x + 4 \end{align*}
satisfied by at least one pair of real numbers $ (x,y)$?
$ \textbf{(A)}\ \text{all } m \qquad \textbf{(B)}\ \text{all } m \neq 0 \qquad \textbf{(C)}\ \text{all } m \neq 1/2 \qquad \textbf{(D)}\ \text{all } m \neq 1 \qquad$
$ \textbf{(E)}\ \text{no values of } m$
TNO 2024 Senior, 6
Let $C$ be a circle, and let $A, B, P$ be three points on $C$. Let $L_A$ and $L_B$ be the tangent lines to $C$ passing through $A$ and $B$, respectively. Let $a$ and $b$ be the distances from $P$ to $L_A$ and $L_B$, respectively, and let $c$ be the distance from $P$ to the chord of $C$ determined by $A$ and $B$. Prove that $c^2 = a \cdot b$.
2013 USA TSTST, 3
Divide the plane into an infinite square grid by drawing all the lines $x=m$ and $y=n$ for $m,n \in \mathbb Z$. Next, if a square's upper-right corner has both coordinates even, color it black; otherwise, color it white (in this way, exactly $1/4$ of the squares are black and no two black squares are adjacent). Let $r$ and $s$ be odd integers, and let $(x,y)$ be a point in the interior of any white square such that $rx-sy$ is irrational. Shoot a laser out of this point with slope $r/s$; lasers pass through white squares and reflect off black squares. Prove that the path of this laser will form a closed loop.
2022 CMIMC Integration Bee, 9
\[\int_e^{e^2} (\log(x))^{\log(x)}(2+\log(\log(x)))\,\mathrm dx\]
[i]Proposed by Connor Gordon[/i]
2015 Regional Olympiad of Mexico Southeast, 3
If $T(n)$ is the numbers of triangles with integers sizes(not congruent with each other) with it´s perimeter is equal to $n$, prove that:
$$T(2012)<T(2015)$$
$$T(2013)=T(2016)$$
PEN O Problems, 6
Let $S$ be a set of integers such that [list][*] there exist $a, b \in S$ with $\gcd(a, b)=\gcd(a-2,b-2)=1$, [*] if $x,y\in S$, then $x^2 -y\in S$.[/list] Prove that $S=\mathbb{Z}$.
2018 Online Math Open Problems, 18
Suppose that $a,b,c$ are real numbers such that $a < b < c$ and $a^3-3a+1=b^3-3b+1=c^3-3c+1=0$. Then $\frac1{a^2+b}+\frac1{b^2+c}+\frac1{c^2+a}$ can be written as $\frac pq$ for relatively prime positive integers $p$ and $q$. Find $100p+q$.
[i]Proposed by Michael Ren[/i]
2004 Harvard-MIT Mathematics Tournament, 10
A floor is tiled with equilateral triangles of side length $1$, as shown. If you drop a needle of length $2$ somewhere on the floor , what is the largest number of triangles it could end up intersecting? (Only count the triangles whose interiors are met by the needle --- touching along edges or at corners doesn't qualify.)
[img]https://cdn.artofproblemsolving.com/attachments/5/6/e7555c22ffe890b46a3ebdbda2169d23e43700.png[/img]
2000 Rioplatense Mathematical Olympiad, Level 3, 1
Let $a$ and $b$ be positive integers such that the number $b^2 + (b +1)^2 +...+ (b + a)^2-3$ is multiple of $5$ and $a + b$ is odd. Calculate the digit of the units of the number $a + b$ written in decimal notation.
2019 Israel National Olympiad, 7
In the plane points $A,B,C$ are marked in blue and points $P,Q$ are marked in red (no 3 marked points lie on a line, and no 4 marked points lie on a circle). A circle is called [b]separating[/b] if all points of one color are inside it, and all points of the other color are outside of it. Denote by $O$ the circumcenter of $ABC$ and by $R$ the circumradius of $ABC$.
Prove that [b]exactly one[/b] of the following holds:
[list]
[*] There exists a separating circle;
[*] There exists a point $X$ on the segment $PQ$ which also lies inside the triangle $ABC$, for which $PX\cdot XQ = R^2-OX^2$.
2005 Postal Coaching, 27
Let $k$ be an even positive integer and define a sequence $<x_n>$ by \[ x_1= 1 , x_{n+1} = k^{x_n} +1. \] Show that $x_n ^2$ divides $x_{n-1}x_{n+1}$ for each $n \geq 2.$