Found problems: 85335
2011 Today's Calculation Of Integral, 675
In the coordinate plane with the origin $O$, consider points $P(t+2,\ 0),\ Q(0, -2t^2-2t+4)\ (t\geq 0).$ If the $y$-coordinate of $Q$ is nonnegative, then find the area of the region swept out by the line segment $PQ$.
[i]2011 Ritsumeikan University entrance exam/Pharmacy[/i]
1993 Czech And Slovak Olympiad IIIA, 1
Find all natural numbers $n$ for which $7^n -1$ is divisible by $6^n -1$
2022 Stanford Mathematics Tournament, 1
If $f(x)=x^4+4x^3+7x^2+6x+2022$, compute $f'(3)$.
2005 Vietnam National Olympiad, 2
Let $(O)$ be a fixed circle with the radius $R$. Let $A$ and $B$ be fixed points in $(O)$ such that $A,B,O$ are not collinear. Consider a variable point $C$ lying on $(O)$ ($C\neq A,B$). Construct two circles $(O_1),(O_2)$ passing through $A,B$ and tangent to $BC,AC$ at $C$, respectively. The circle $(O_1)$ intersects the circle $(O_2)$ in $D$ ($D\neq C$). Prove that:
a) \[ CD\leq R \]
b) The line $CD$ passes through a point independent of $C$ (i.e. there exists a fixed point on the line $CD$ when $C$ lies on $(O)$).
1990 Irish Math Olympiad, 4
The real number $x$ satisfies all the inequalities $$2^k<x^k+x^{k+1}<2^{k+1}$$ for $k=1,2,\dots ,n$. What is the greatest possible value of $n$?
2018 239 Open Mathematical Olympiad, 10-11.6
For which positive integers $n$, $m$ does there exist a polynomial of degree $n$, all coefficients of which are powers of $m$ with integer exponents, having $n$ rational roots, counting multiplicities?
[i]Proposed by Fedor Petrov[/i]
2005 Serbia Team Selection Test, 2
$$problem2$$:Determine the number of 100-digit numbers whose all digits are odd, and in
which every two consecutive digits differ by 2
1990 All Soviet Union Mathematical Olympiad, 529
A quadratic polynomial $p(x)$ has positive real coefficients with sum $1$. Show that given any positive real numbers with product $1$, the product of their values under $p$ is at least $1$.
2014 AMC 10, 22
Eight semicircles line the inside of a square with side length 2 as shown. What is the radius of the circle tangent to all of these semicircles?
[asy]
scale(200);
draw(scale(.5)*((-1,-1)--(1,-1)--(1,1)--(-1,1)--cycle));
path p = arc((.25,-.5),.25,0,180)--arc((-.25,-.5),.25,0,180);
draw(p);
p=rotate(90)*p; draw(p);
p=rotate(90)*p; draw(p);
p=rotate(90)*p; draw(p);
draw(scale((sqrt(5)-1)/4)*unitcircle);
[/asy]
$\text{(A) } \dfrac{1+\sqrt2}4 \quad \text{(B) } \dfrac{\sqrt5-1}2 \quad \text{(C) } \dfrac{\sqrt3+1}4 \quad \text{(D) } \dfrac{2\sqrt3}5 \quad \text{(E) } \dfrac{\sqrt5}3$
2025 PErA, P5
We have an $n \times n$ board, filled with $n$ rectangles aligned to the grid. The $n$ rectangles cover all the board and are never superposed. Find, in terms of $n$, the smallest value the sum of the $n$ diagonals of the rectangles can take.
2015 Harvard-MIT Mathematics Tournament, 8
Find the number of ordered pairs of integers $(a,b)\in\{1,2,\ldots,35\}^2$ (not necessarily distinct) such that $ax+b$ is a "quadratic residue modulo $x^2+1$ and $35$", i.e. there exists a polynomial $f(x)$ with integer coefficients such that either of the following $\textit{equivalent}$ conditions holds:
[list]
[*] there exist polynomials $P$, $Q$ with integer coefficients such that $f(x)^2-(ax+b)=(x^2+1)P(x)+35Q(x)$;
[*] or more conceptually, the remainder when (the polynomial) $f(x)^2-(ax+b)$ is divided by (the polynomial) $x^2+1$ is a polynomial with integer coefficients all divisible by $35$.
[/list]
1967 IMO Longlists, 48
Determine all positive roots of the equation $ x^x = \frac{1}{\sqrt{2}}.$
2015 NZMOC Camp Selection Problems, 8
Determine all positive integers $n$ which have a divisor $d$ with the property that $dn + 1$ is a divisor of $d^2 + n^2$.
2001 IMO Shortlist, 6
Let $ABC$ be a triangle and $P$ an exterior point in the plane of the triangle. Suppose the lines $AP$, $BP$, $CP$ meet the sides $BC$, $CA$, $AB$ (or extensions thereof) in $D$, $E$, $F$, respectively. Suppose further that the areas of triangles $PBD$, $PCE$, $PAF$ are all equal. Prove that each of these areas is equal to the area of triangle $ABC$ itself.
2015 Math Prize for Girls Problems, 13
Joel selected an acute angle $x$ (strictly between 0 and 90 degrees) and wrote the values of $\sin x$, $\cos x$, and $\tan x$ on three different cards. Then he gave those cards to three students, Malvina, Paulina, and Georgina, one card to each, and asked them to figure out which trigonometric function (sin, cos, or tan) produced their cards. Even after sharing the values on their cards with each other, only Malvina was able to surely identify which function produced the value on her card. Compute the sum of all possible values that Joel wrote on Malvina's card.
2022 AMC 10, 8
Consider the following $100$ sets of $10$ elements each:
\begin{align*}
&\{1,2,3,\cdots,10\}, \\
&\{11,12,13,\cdots,20\},\\
&\{21,22,23,\cdots,30\},\\
&\vdots\\
&\{991,992,993,\cdots,1000\}.
\end{align*}
How many of these sets contain exactly two multiples of $7$?
$\textbf{(A)} 40\qquad\textbf{(B)} 42\qquad\textbf{(C)} 43\qquad\textbf{(D)} 49\qquad\textbf{(E)} 50$
2005 Today's Calculation Of Integral, 58
Let $f(x)=\frac{e^x}{e^x+1}$
Prove the following equation.
\[\int_a^b f(x)dx+\int_{f(a)}^{f(b)} f^{-1}(x)dx=bf(b)-af(a)\]
2016 Junior Balkan Team Selection Test, 1
Let rightangled $\triangle ABC$ be given with right angle at vertex $C$. Let $D$ be foot of altitude from $C$ and let $k$ be circle that touches $BD$ at $E$, $CD$ at $F$ and circumcircle of $\triangle ABC$ at $G$.
$a.)$ Prove that points $A$, $F$ and $G$ are collinear.
$b.)$ Express radius of circle $k$ in terms of sides of $\triangle ABC$.
2017 Pan African, Problem 1
We consider the real sequence $(x_n)$ defined by $x_0=0, x_1=1$ and $x_{n+2}=3x_{n+1}-2x_n$ for $n=0,1,...$
We define the sequence $(y_n)$ by $y_n=x_n^2+2^{n+2}$ for every non negative integer $n$.
Prove that for every $n>0$, $y_n$ is the square of an odd integer
2006 Princeton University Math Competition, 8
Given that triangle $ABC$ has side lengths $a=7$, $b=8$ , $c=5$, find $$(\sin (A)+\sin (B)+\sin (C)) \cdot \left(\cot \frac{A}{2}+\cot \frac{B}{2}+\cot \frac{C}{2}\right).$$
.
2004 Spain Mathematical Olympiad, Problem 2
${ABCD}$ is a quadrilateral, ${P}$ and ${Q}$ are midpoints of the diagonals ${BD}$ and ${AC}$, respectively. The lines parallel to the diagonals originating from ${P}$ and ${Q}$ intersect in the point ${O}$. If we join the four midpoints of the sides, ${X}$, ${Y}$, ${Z}$, and ${T}$, to ${O}$, we form four quadrilaterals: ${OXBY}$, ${OYCZ}$, ${OZDT}$, and ${OTAX}$. Prove that the four newly formed quadrilaterals have the same areas.
2024 Singapore Junior Maths Olympiad, Q1
Let $ABC$ be an isosceles right-angled triangle of area 1. Find the length of the shortest segment that divides the triangle into 2 parts of equal area.
2014 Chile TST Ibero, 3
Let $x_0 = 5$ and define the sequence recursively as $x_{n+1} = x_n + \frac{1}{x_n}$. Prove that:
\[
45 < x_{1000} < 45.1.
\]
2013 Putnam, 3
Suppose that the real numbers $a_0,a_1,\dots,a_n$ and $x,$ with $0<x<1,$ satisfy \[\frac{a_0}{1-x}+\frac{a_1}{1-x^2}+\cdots+\frac{a_n}{1-x^{n+1}}=0.\] Prove that there exists a real number $y$ with $0<y<1$ such that \[a_0+a_1y+\cdots+a_ny^n=0.\]
2011 India IMO Training Camp, 1
Let $ABC$ be a triangle each of whose angles is greater than $30^{\circ}$. Suppose a circle centered with $P$ cuts segments $BC$ in $T,Q; CA$ in $K,L$ and $AB$ in $M,N$ such that they are on a circle in counterclockwise direction in that order.Suppose further $PQK,PLM,PNT$ are equilateral. Prove that:
$a)$ The radius of the circle is $\frac{2abc}{a^2+b^2+c^2+4\sqrt{3}S}$ where $S$ is area.
$b) a\cdot AP=b\cdot BP=c\cdot PC.$