Found problems: 85335
1993 AMC 12/AHSME, 10
Let $r$ be the number that results when both the base and the exponent of $a^b$ are tripled, where $a, b>0$. If $r$ equals the product of $a^b$ and $x^b$ where $x>0$, then $x=$
$ \textbf{(A)}\ 3 \qquad\textbf{(B)}\ 3a^2 \qquad\textbf{(C)}\ 27a^2 \qquad\textbf{(D)}\ 2a^{3b} \qquad\textbf{(E)}\ 3a^{2b} $
2014 Puerto Rico Team Selection Test, 1
Let $ABCD$ be a parallelogram with $AB>BC$ and $\angle DAB$ less than $\angle ABC$. The perpendicular bisectors of sides $AB$ and $BC$ intersect at the point $M$ lying on the extension of $AD$. If $\angle MCD=15^{\circ}$, find the measure of $\angle ABC$
2020 JBMO Shortlist, 1
Let $\triangle ABC$ be an acute triangle. The line through $A$ perpendicular to $BC$ intersects $BC$ at $D$.
Let $E$ be the midpoint of $AD$ and $\omega$ the the circle with center $E$ and radius equal to $AE$. The line
$BE$ intersects $\omega$ at a point $X$ such that $X$ and $B$ are not on the same side of $AD$ and the line $CE$
intersects $\omega$ at a point $Y$ such that $C$ and $Y$ are not on the same side of $AD$. If both of the intersection
points of the circumcircles of $\triangle BDX$ and $\triangle CDY$ lie on the line $AD$, prove that $AB = AC$.
2010 Cuba MO, 8
Let $ABCDE$ be a convex pentagon that has $AB < BC$, $AE <ED$ and $AB + CD + EA = BC + DE$. Variable points $F,G$ and $H$ are taken that move on the segments $BC$, $CD$ and $OF$ respectively . $B'$ is defined as the projection of $B$ on $AF$, $C'$ as the projection of $C$ on $FG$, $D'$ as the projection of $D$ on $GH$ and $E'$ as the projection of $E$ onto $HA$. Prove that there is at least one quadrilateral $B'C'D'E'$ when $F,G$ and $H$ move on their sides, which is a parallelogram.
2012 Pre-Preparation Course Examination, 5
The $2^{nd}$ order differentiable function $f:\mathbb R \longrightarrow \mathbb R$ is in such a way that for every $x\in \mathbb R$ we have $f''(x)+f(x)=0$.
[b]a)[/b] Prove that if in addition, $f(0)=f'(0)=0$, then $f\equiv 0$.
[b]b)[/b] Use the previous part to show that there exist $a,b\in \mathbb R$ such that $f(x)=a\sin x+b\cos x$.
1996 Singapore Team Selection Test, 2
Let $ k$ be a positive integer. Show that there are infinitely many perfect squares of the form $ n \cdot 2^k \minus{} 7$ where $ n$ is a positive integer.
2020 LMT Fall, A11 B20
Two sequences of nonzero reals $a_1, a_2, a_3, \dots$ and $b_2, b_3, \dots$ are such that $b_n=\prod_{i=1}^{n} a_i$ and $a_n=\frac{b_n^2}{3b_n-3}$ for all integers $n > 1$. Given that $a_1=\frac{1}{2}$, find $\lvert b_{60}\rvert$.
[i]Proposed by Andrew Zhao[/i]
2022 Novosibirsk Oral Olympiad in Geometry, 7
Vera has several identical matches, from which she makes a triangle. Vera wants any two sides of this triangle to differ in length by at least $10$ matches, but it turned out that it is impossible to add such a triangle from the available matches (it is impossible to leave extra matches). What is the maximum number of matches Vera can have?
2003 Cuba MO, 2
Let $A$ be a point outside the circle $\omega$ . The tangents from $A$ touch the circle at $B$ and $C$. Let $P$ be an arbitrary point on extension of $AC$ towards $C$, $Q$ the projection of $C$ onto $PB$ and $E$ the second intersection point of the circumcircle of $ABP$ with the circle $\omega$ . Prove that $\angle PEQ = 2\angle APB$
2008 Junior Balkan Team Selection Tests - Moldova, 2
[b]BJ2. [/b] Positive real numbers $ a,b,c$ satisfy inequality $ \frac {3}{2}\geq a \plus{} b \plus{} c$. Find the smallest possible value for
$ S \equal{} abc \plus{} \frac {1}{abc}$
2019 HMNT, 8
Omkar, Krit1, Krit2, and Krit3 are sharing $x > 0$ pints of soup for dinner. Omkar always takes $1$ pint of soup (unless the amount left is less than one pint, in which case he simply takes all the remaining soup). Krit1 always takes $\frac16$ of what is left, Krit2 always takes $\frac15$ of what is left, and Krit3 always takes $\frac14$ of what is left. They take soup in the order of Omkar, Krit1, Krit2, Krit3, and then cycle through this order until no soup remains. Find all $x$ for which everyone gets the same amount of soup.
2013 JBMO Shortlist, 5
A circle passing through the midpoint $M$ of the side $BC$ and the vertex $A$ of the triangle $ABC$ intersects the segments $AB$ and $AC$ for the second time in the points $P$ and $Q$, respectively. Prove that if $\angle BAC=60^{\circ}$, then $AP+AQ+PQ<AB+AC+\frac{1}{2} BC$.
2010 Indonesia TST, 1
Given $ a,b, c $ positive real numbers satisfying $ a+b+c=1 $.
Prove that \[ \dfrac{1}{\sqrt{ab+bc+ca}}\ge \sqrt{\dfrac{2a}{3(b+c)}} +\sqrt{\dfrac{2b}{3(c+a)}} + \sqrt{\dfrac{2c}{3(a+b)}} \ge \sqrt{a} +\sqrt{b}+\sqrt{c} \]
1940 Putnam, A1
Prove that if $f(x)$ is a polynomial with integer coefficients and there exists an integer $k$ such that none of $f(1),\ldots,f(k)$ is divisible by $k$, then $f(x)$ has no integral root.
2008 AMC 12/AHSME, 19
A function $ f$ is defined by $ f(z) \equal{} (4 \plus{} i) z^2 \plus{} \alpha z \plus{} \gamma$ for all complex numbers $ z$, where $ \alpha$ and $ \gamma$ are complex numbers and $ i^2 \equal{} \minus{} 1$. Suppose that $ f(1)$ and $ f(i)$ are both real. What is the smallest possible value of $ | \alpha | \plus{} |\gamma |$?
$ \textbf{(A)} \; 1 \qquad \textbf{(B)} \; \sqrt {2} \qquad \textbf{(C)} \; 2 \qquad \textbf{(D)} \; 2 \sqrt {2} \qquad \textbf{(E)} \; 4 \qquad$
2010 Math Prize For Girls Problems, 7
The graph of ${(x^2 + y^2 - 1)}^3 = x^2 y^3$ is a heart-shaped curve, shown in the figure below.
[asy]
import graph;
unitsize(10);
real f(real x)
{
return sqrt(cbrt(x^4) - 4 x^2 + 4);
}
real g(real x)
{
return (cbrt(x^2) + f(x))/2;
}
real h(real x)
{
return (cbrt(x^2) - f(x)) / 2;
}
real xmax = 1.139028;
draw(graph(g, -xmax, xmax) -- reverse(graph(h, -xmax, xmax)) -- cycle);
xaxis("$x$", -1.5, 1.5, above = true);
yaxis("$y$", -1.5, 1.5, above = true);
[/asy]
For how many ordered pairs of integers $(x, y)$ is the point $(x, y)$ inside or on this curve?
1960 AMC 12/AHSME, 22
The eqquality $(x+m)^2-(x+n)^2=(m-n)^2$, where $m$ and $n$ are [i]unequal[/i] non-zero constants, is satisfied by $x=am+bn$, where:
$ \textbf{(A)}\ a = 0, b \text{ } \text{has a unique non-zero value}\qquad$
$\textbf{(B)}\ a = 0, b \text{ } \text{has two non-zero values}\qquad$
$\textbf{(C)}\ b = 0, a \text{ } \text{has a unique non-zero value}\qquad$
$\textbf{(D)}\ b = 0, a \text{ } \text{has two non-zero values}\qquad$
$\textbf{(E)}\ a \text{ } \text{and} \text{ } b \text{ } \text{each have a unique non-zero value} $
2021-IMOC, C9
In a simple graph, there exist two vertices $A,B$ such that there are exactly $100$ shortest paths from $A$ to $B$. Find the minimum number of edges in the graph.
[i]CSJL[/i]
2014 Middle European Mathematical Olympiad, 3
Let $K$ and $L$ be positive integers. On a board consisting of $2K \times 2L$ unit squares an ant starts in the lower left corner square and walks to the upper right corner square. In each step it goes horizontally or vertically to a neighbouring square. It never visits a square twice. At the end some squares may remain unvisited.
In some cases the collection of all unvisited squares forms a single rectangle. In such cases, we call this rectangle [i]MEMOrable[/i].
Determine the number of different MEMOrable rectangles.
[i]Remark: Rectangles are different unless they consist of exactly the same squares.[/i]
1993 All-Russian Olympiad, 3
Find all functions $f(x)$ with the domain of all positive real numbers, such that for any positive numbers $x$ and $y$, we have $f(x^y)=f(x)^{f(y)}$.
2025 International Zhautykov Olympiad, 5
Let $A_1C_2B_1B_2C_1A_2$ be a cyclic convex hexagon inscribed in circle $\Omega$, centered at $O$. Let $\{ P \} = A_2B_2 \cap A_1B_1$ and $\{ Q \} = A_2C_2 \cap A_1C_1$. Let $\Gamma_1$ be a circle tangent to $OB_1$ and $OC_1$ at $B_1,C_1$ respectively. Similarly, define $\Gamma_2$ to be the circle tangent to $OB_2,OC_2$ at $B_2, C_2$ respectively. Prove that there is a homothety that sends $\Gamma_1$ to $\Gamma_2$, whose center lies on $PQ$
1965 Dutch Mathematical Olympiad, 2
Prove that $S_1 = (n + 1)^2 + (n + 2)^2 +...+ (n + 5)^2$ is divisible by $5$ for every $n$.
Prove that for no $n$: $\sum_{\ell=1}^5 (n+\ell)^2$ is a perfect square.
Let $S_2=(n + 6)^2 + (n + 7)^2 + ... + (n + 10)^2$. Prove that $S_1 \cdot S_2$ is divisible by $150$.
1993 AIME Problems, 3
The table below displays some of the results of last summer's Frostbite Falls Fishing Festival, showing how many contestants caught $n$ fish for various values of $n$.
\[ \begin{array}{|c|c|c|c|c|c|c|c|c|} \hline n & 0 & 1 & 2 & 3 & \dots & 13 & 14 & 15 \\ \hline \text{number of contestants who caught} \ n \ \text{fish} & 9 & 5 & 7 & 23 & \dots & 5 & 2 & 1 \\ \hline \end{array} \]
In the newspaper story covering the event, it was reported that
(a) the winner caught 15 fish;
(b) those who caught 3 or more fish averaged 6 fish each;
(c) those who caught 12 or fewer fish averaged 5 fish each.
What was the total number of fish caught during the festival?
2007 All-Russian Olympiad Regional Round, 8.6
A number $ B$ is obtained from a positive integer number $ A$ by permuting its decimal digits. The number $ A\minus{}B\equal{}11...1$ ($ n$ of $ 1's$). Find the smallest possible positive value of $ n$.
2014 Contests, 1
Consider the number $\left(101^2-100^2\right)\cdot\left(102^2-101^2\right)\cdot\left(103^2-102^2\right)\cdot...\cdot\left(200^2-199^2\right)$.
[list=a]
[*] Determine its units digit.
[*] Determine its tens digit.
[/list]