Found problems: 85335
2022 BMT, Tie 2
Call a positive whole number [i]rickety [/i] if it is three times the product of its digits. There are two $2$-digit numbers that are rickety. What is their sum?
1957 AMC 12/AHSME, 49
The parallel sides of a trapezoid are $ 3$ and $ 9$. The non-parallel sides are $ 4$ and $ 6$. A line parallel to the bases divides the trapezoid into two trapezoids of equal perimeters. The ratio in which each of the non-parallel sides is divided is:
[asy]defaultpen(linewidth(.8pt));
unitsize(2cm);
pair A = origin;
pair B = (2.25,0);
pair C = (2,1);
pair D = (1,1);
pair E = waypoint(A--D,0.25);
pair F = waypoint(B--C,0.25);
draw(A--B--C--D--cycle);
draw(E--F);
label("6",midpoint(A--D),NW);
label("3",midpoint(C--D),N);
label("4",midpoint(C--B),NE);
label("9",midpoint(A--B),S);[/asy]$ \textbf{(A)}\ 4: 3\qquad \textbf{(B)}\ 3: 2\qquad \textbf{(C)}\ 4: 1\qquad \textbf{(D)}\ 3: 1\qquad \textbf{(E)}\ 6: 1$
2007 Federal Competition For Advanced Students, Part 2, 3
Determine all rhombuses $ ABCD$ with the given length $ 2a$ of ist sides by giving the angle $ \alpha \equal{} \angle BAD$, such that there exists a circle which cuts each side of the rhombus in a chord of length $ a$.
2004 Denmark MO - Mohr Contest, 3
The digits from $1$ to $9$ are placed in the figure below with one digit in each square. The sum of three numbers placed in the same horizontal or vertical line is $13$. Show that the marked place says $4$.
[img]https://cdn.artofproblemsolving.com/attachments/a/f/517b644caf59bbc57701662f21d57465855dc1.png[/img]
2014 Online Math Open Problems, 1
In English class, you have discovered a mysterious phenomenon -- if you spend $n$ hours on an essay, your score on the essay will be $100\left( 1-4^{-n} \right)$ points if $2n$ is an integer, and $0$ otherwise. For example, if you spend $30$ minutes on an essay you will get a score of $50$, but if you spend $35$ minutes on the essay you somehow do not earn any points.
It is 4AM, your English class starts at 8:05AM the same day, and you have four essays due at the start of class. If you can only work on one essay at a time, what is the maximum possible average of your essay scores?
[i]Proposed by Evan Chen[/i]
2019 Sharygin Geometry Olympiad, 16
Let $AH_1$ and $BH_2$ be the altitudes of triangle $ABC$. Let the tangent to the circumcircle of $ABC$ at $A$ meet $BC$ at point $S_1$, and the tangent at $B$ meet $AC$ at point $S_2$. Let $T_1$ and $T_2$ be the midpoints of $AS_1$ and $BS_2$ respectively. Prove that $T_1T_2$, $AB$ and $H_1H_2$ concur.
1991 Federal Competition For Advanced Students, 4
Let $ AB$ be a chord of a circle $ k$ of radius $ r$, with $ AB\equal{}c$.
$ (a)$ Construct the triangle $ ABC$ with $ C$ on $ k$ in which a median from $ A$ or $ B$ is of a given length $ d.$
$ (b)$ For which $ c$ and $ d$ is this triangle unique?
2012 All-Russian Olympiad, 2
The points $A_1,B_1,C_1$ lie on the sides sides $BC,AC$ and $AB$ of the triangle $ABC$ respectively. Suppose that $AB_1-AC_1=CA_1-CB_1=BC_1-BA_1$. Let $I_A, I_B, I_C$ be the incentres of triangles $AB_1C_1,A_1BC_1$ and $A_1B_1C$ respectively. Prove that the circumcentre of triangle $I_AI_BI_C$ is the incentre of triangle $ABC$.
2002 AMC 8, 15
Which of the following polygons has the largest area?
[asy]
size(330);
int i,j,k;
for(i=0;i<5; i=i+1) {
for(j=0;j<5;j=j+1) {
for(k=0;k<5;k=k+1) {
dot((6i+j, k));
}}}
draw((0,0)--(4,0)--(3,1)--(3,3)--(2,3)--(2,1)--(1,1)--cycle);
draw(shift(6,0)*((0,0)--(4,0)--(4,1)--(3,1)--(3,2)--(2,1)--(1,1)--(0,2)--cycle));
draw(shift(12,0)*((0,1)--(1,0)--(3,2)--(3,3)--(1,1)--(1,3)--(0,4)--cycle));
draw(shift(18,0)*((0,1)--(2,1)--(3,0)--(3,3)--(2,2)--(1,3)--(1,2)--(0,2)--cycle));
draw(shift(24,0)*((1,0)--(2,1)--(2,3)--(3,2)--(3,4)--(0,4)--(1,3)--cycle));
label("$A$", (0*6+2, 0), S);
label("$B$", (1*6+2, 0), S);
label("$C$", (2*6+2, 0), S);
label("$D$", (3*6+2, 0), S);
label("$E$", (4*6+2, 0), S);
[/asy]
$ \textbf{(A)}\text{A}\qquad\textbf{(B)}\ \text{B}\qquad\textbf{(C)}\ \text{C}\qquad\textbf{(D)}\ \text{D}\qquad\textbf{(E)}\ \text{E} $
2011 Mathcenter Contest + Longlist, 4 sl4
At the $69$ Thailand-Yaranaikian meeting attended by $96$ Thai delegates and a number (unknown) from the Yaranakian country. Some time after the meeting took place, the meeting also discovered something amazing that happened in this meeting!! That is, regardless of whether we select at least $69$ of Thai participants and select all the Yaranikian country participants who are known to Thais in the initial selection group, there is at least $1$ person fo form a minority. They found in that minority, there was always $1$ more Yaranikhians than Thais. Prove that there must be at least $28$ of the Yaranaikian attendees who know the Thai delegates.
(Note: In this meeting, none of the attendees were half-breeds. Thai-Yara Nikian)
[i](tatari/nightmare)[/i]
2013 IPhOO, 6
A fancy bathroom scale is calibrated in Newtons. This scale is put on a ramp, which is at a $40^\circ$ angle to the horizontal. A box is then put on the scale and the box-scale system is then pushed up the ramp by a horizontal force $F$. The system slides up the ramp at a constant speed. If the bathroom scale reads $R$ and the coefficient of static friction between the system and the ramp is $0.40$, what is $\frac{F}{R}$? Round to the nearest thousandth.
[i](Proposed by Ahaan Rungta)[/i]
2005 National High School Mathematics League, 10
In tetrahedron $ABCD$, the volume of tetrahedron $ABCD$ is $\frac{1}{6}$, and $\angle ACB=45^{\circ},AD+BC+\frac{AC}{\sqrt2}=3$, then $CD=$________.
1951 Moscow Mathematical Olympiad, 202
Dividing $x^{1951} - 1$ by $P(x) = x^4 + x^3 + 2x^2 + x + 1$ one gets a quotient and a remainder. Find the coefficient of $x^{14}$ in the quotient.
2008 Dutch IMO TST, 5
Let $\vartriangle ABC$ be a right triangle with $\angle B = 90^o$ and $|AB| > |BC|$, and let $\Gamma$ be the semicircle with diameter $AB$ that lies on the same side as $C$. Let $P$ be a point on $\Gamma$ such that $|BP| = |BC|$ and let $Q$ be on $AB$ such that $|AP| = |AQ|$. Prove that the midpoint of $CQ$ lies on $\Gamma$.
1952 AMC 12/AHSME, 30
When the sum of the first ten terms of an arithmetic progression is four times the sum of the first five terms, the ratio of the first term to the common difference is:
$ \textbf{(A)}\ 1: 2 \qquad\textbf{(B)}\ 2: 1 \qquad\textbf{(C)}\ 1: 4 \qquad\textbf{(D)}\ 4: 1 \qquad\textbf{(E)}\ 1: 1$
2009 IMO Shortlist, 6
Let the sides $AD$ and $BC$ of the quadrilateral $ABCD$ (such that $AB$ is not parallel to $CD$) intersect at point $P$. Points $O_1$ and $O_2$ are circumcenters and points $H_1$ and $H_2$ are orthocenters of triangles $ABP$ and $CDP$, respectively. Denote the midpoints of segments $O_1H_1$ and $O_2H_2$ by $E_1$ and $E_2$, respectively. Prove that the perpendicular from $E_1$ on $CD$, the perpendicular from $E_2$ on $AB$ and the lines $H_1H_2$ are concurrent.
[i]Proposed by Eugene Bilopitov, Ukraine[/i]
2009 Denmark MO - Mohr Contest, 1
In the figure, triangle $ADE$ is produced from triangle $ABC$ by a rotation by $90^o$ about the point $A$. If angle $D$ is $60^o$ and angle $E$ is $40^o$, how large is then angle $u$?
[img]https://1.bp.blogspot.com/-6Fq2WUcP-IA/Xzb9G7-H8jI/AAAAAAAAMWY/hfMEAQIsfTYVTdpd1Hfx15QPxHmfDLEkgCLcBGAsYHQ/s0/2009%2BMohr%2Bp1.png[/img]
IV Soros Olympiad 1997 - 98 (Russia), 10.1
Two sides of the cyclic quadrilateral $ABCD$ are known: $AB = a$, $BC = b$. A point $K$ is taken on the side $CD$ so that $CK = m$. A circle passing through $B$, $K$ and $D$ intersects line $DA$ at a point $M$, different from $D$. Find $AM$.
2007 ITest, 15
Form a pentagon by taking a square of side length $1$ and an equilateral triangle of side length $1$ and placing the triangle so that one of its sides coincides with a side of the square. Then "circumscribe" a circle around the pentagon, passing through three of its vertices, so that the circle passes through exactly one vertex of the equilateral triangle, and exactly two vertices of the square. What is the radius of the circle?
$\textbf{(A) }\dfrac23\hspace{14.4em}\textbf{(B) }\dfrac34\hspace{14.4em}\textbf{(C) }1$
$\textbf{(D) }\dfrac54\hspace{14.4em}\textbf{(E) }\dfrac43\hspace{14.4em}\textbf{(F) }\dfrac{\sqrt2}2$
$\textbf{(G) }\dfrac{\sqrt3}2\hspace{13.5em}\textbf{(H) }\sqrt2\hspace{13.8em}\textbf{(I) }\sqrt3$
$\textbf{(J) }\dfrac{1+\sqrt3}2\hspace{12em}\textbf{(K) }\dfrac{2+\sqrt6}2\hspace{11.9em}\textbf{(L) }\dfrac76$
$\textbf{(M) }\dfrac{2+\sqrt6}4\hspace{11.5em}\textbf{(N) }\dfrac45\hspace{14.4em}\textbf{(O) }2007$
2012 India IMO Training Camp, 3
Let $\mathbb{R}^{+}$ denote the set of all positive real numbers. Find all functions $f:\mathbb{R}^{+}\longrightarrow \mathbb{R}$ satisfying
\[f(x)+f(y)\le \frac{f(x+y)}{2}, \frac{f(x)}{x}+\frac{f(y)}{y}\ge \frac{f(x+y)}{x+y},\]
for all $x, y\in \mathbb{R}^{+}$.
2004 AMC 12/AHSME, 8
A grocer makes a display of cans in which the top row has one can and each lower row has two more cans than the row above it. If the display contains $ 100$ cans, how many rows does it contain?
$ \textbf{(A)}\ 5\qquad
\textbf{(B)}\ 8\qquad
\textbf{(C)}\ 9\qquad
\textbf{(D)}\ 10\qquad
\textbf{(E)}\ 11$
2004 Dutch Mathematical Olympiad, 4
Two circles $C_1$ and $C_2$ touch each other externally in a point $P$. At point $C_1$ there is a point $Q$ such that the tangent line in $Q$ at $C_1$ intersects the circle $C_2$ at points $A$ and $B$. The line $QP$ still intersects $C_2$ at point $C$.
Prove that triangle $ABC$ is isosceles.
2023 Costa Rica - Final Round, 3.1
Let $\mathbb Z^{\geq 0}$ be the set of all non-negative integers. Consider a function $f:\mathbb Z^{\geq 0} \to \mathbb Z^{\geq 0}$ such that $f(0)=1$ and $f(1)=1$, and that for any integer $n \geq 1$, we have
\[f(n + 1)f(n - 1) = nf(n)f(n - 1) + (f(n))^2.\]
Determine the value of $f(2023)/f(2022)$.
2024 Putnam, B3
Let $r_n$ be the $n$th smallest positive solution to $\tan x=x$, where the argument of tangent is in radians. Prove that
\[
0<r_{n+1}-r_n-\pi<\frac{1}{(n^2+n)\pi}
\]
for $n\geq 1$.
1974 Bulgaria National Olympiad, Problem 3
(a) Find all real numbers $p$ for which the inequality
$$x_1^2+x_2^2+x_3^2\ge p(x_1x_2+x_2x_3)$$
is true for all real numbers $x_1,x_2,x_3$.
(b) Find all real numbers $q$ for which the inequality
$$x_1^2+x_2^2+x_3^2+x_4^2\ge q(x_1x_2+x_2x_3+x_3x_4)$$
is true for all real numbers $x_1,x_2,x_3,x_4$.
[i]I. Tonov[/i]