Found problems: 85335
2014 Grand Duchy of Lithuania, 1
Determine all functions $f : R \to R$ such that $f(xy + f(x)) = xf(y) + f(x)$ holds for any $x, y \in R$.
2019 Korea National Olympiad, 5
Find all functions $f$ such that $f:\mathbb{R}\rightarrow \mathbb{R}$ and $f(f(x)-x+y^2)=yf(y)$
2016 Stars of Mathematics, 1
Find the minimum number of perfect cubes such that their sum is equal to $ 346^{346} . $
2013 AMC 8, 23
Angle $ABC$ of $\triangle ABC$ is a right angle. The sides of $\triangle ABC$ are the diameters of semicircles as shown. The area of the semicircle on $\overline{AB}$ equals $8\pi$, and the arc of the semicircle on $\overline{AC}$ has length $8.5\pi$. What is the radius of the semicircle on $\overline{BC}$?
[asy]
import graph;
draw((0,8)..(-4,4)..(0,0)--(0,8));
draw((0,0)..(7.5,-7.5)..(15,0)--(0,0));
real theta = aTan(8/15);
draw(arc((15/2,4),17/2,-theta,180-theta));
draw((0,8)--(15,0));
label("$A$", (0,8), NW);
label("$B$", (0,0), SW);
label("$C$", (15,0), SE);[/asy]
$\textbf{(A)}\ 7 \qquad \textbf{(B)}\ 7.5 \qquad \textbf{(C)}\ 8 \qquad \textbf{(D)}\ 8.5 \qquad \textbf{(E)}\ 9$
2010 Dutch IMO TST, 2
Let $A$ and $B$ be positive integers. De fine the arithmetic sequence $a_0, a_1, a_2, ...$ by $a_n = A_n + B$. Suppose that there exists an $n\ge 0$ such that $a_n$ is a square. Let $M$ be a positive integer such that $M^2$ is the smallest square in the sequence. Prove that $M < A +\sqrt{B}$.
2016 Costa Rica - Final Round, G2
Let $ABCD$ be a convex quadrilateral, such that $ A$, $ B$, $C$, and $D$ lie on a circle, with $\angle DAB < \angle ABC$. Let $I$ be the intersection of the bisector of $\angle ABC$ with the bisector of $\angle BAD$. Let $\ell$ be the parallel line to $CD$ passing through point $I$. Suppose $\ell$ cuts segments $DA$ and $BC$ at $ L$ and $J$, respectively. Prove that $AL + JB = LJ$.
1978 Germany Team Selection Test, 2
Let $S$ be a convex quadrilateral $ABCD$ and $O$ a point inside it. The feet of the perpendiculars from $O$ to $AB, BC, CD, DA$ are $A_1, B_1, C_1, D_1$ respectively. The feet of the perpendiculars from $O$ to the sides of $S_i$, the quadrilateral $A_iB_iC_iD_i$, are $A_{i+1}B_{i+1}C_{i+1}D_{i+1}$, where $i = 1, 2, 3.$ Prove that $S_4$ is similar to S.
1985 IMO Longlists, 12
Find the maximum value of
\[\sin^2 \theta_1+\sin^2 \theta_2+\cdots+\sin^2 \theta_n\]
subject to the restrictions $0 \leq \theta_i , \theta_1+\theta_2+\cdots+\theta_n=\pi.$
2010 HMNT, 10
You are given two diameters $AB$ and $CD$ of circle $\Omega$ with radius $1$. A circle is drawn in one of the smaller sectors formed such that it is tangent to $AB$ at $E$, tangent to $CD$ at $F$, and tangent to $\Omega$ at $P$. Lines $PE$ and $PF$ intersect $\Omega$ again at $X$ and $Y$ . What is the length of $XY$ , given that $AC = \frac23$ ?
2010 Princeton University Math Competition, 6
All the diagonals of a regular decagon are drawn. A regular decagon satisfies the property that if three diagonals concur, then one of the three diagonals is a diameter of the circumcircle of the decagon. How many distinct intersection points of diagonals are in the interior of the decagon?
2021 Pan-African, 5
Find all functions $f$ $:$ $\mathbb{R} \rightarrow \mathbb{R}$ such that $\forall x,y \in \mathbb{R}$ :
$$(f(x)+y)(f(y)+x)=f(x^2)+f(y^2)+2f(xy)$$
2019 Durer Math Competition Finals, 3
Let $P$ be an interior point of triangle $ABC$. The lines $AP$, $BP$ and $CP$ divide each of the three sides into two segments. If the so-obtained six segments all have distinct integer lengths, what is the minimum possible perimeter of $ABC$?
2011 USA Team Selection Test, 5
Let $c_n$ be a sequence which is defined recursively as follows: $c_0 = 1$, $c_{2n+1} = c_n$ for $n \geq 0$, and $c_{2n} = c_n + c_{n-2^e}$ for $n > 0$ where $e$ is the maximal nonnegative integer such that $2^e$ divides $n$. Prove that
\[\sum_{i=0}^{2^n-1} c_i = \frac{1}{n+2} {2n+2 \choose n+1}.\]
1982 Bundeswettbewerb Mathematik, 2
Decide whether every triangle $ABC$ in space can be orthogonally projected onto a plane such that the projection is an equilateral triangle $A'B'C'$.
2024 Princeton University Math Competition, B2
Alien Tanvi has a favorite number, but somehow she’s managed to forget it. She remembers that it can be written as $x^2+\tfrac{1}{x^2},$ where $x$ is a real number satisfying $x^4+4x^2+\tfrac{4}{x^2}+\tfrac{1}{x^4}=523.$ What is Alien Tanvi's favorite number?
2012 Belarus Team Selection Test, 3
For each positive integer $k,$ let $t(k)$ be the largest odd divisor of $k.$ Determine all positive integers $a$ for which there exists a positive integer $n,$ such that all the differences
\[t(n+a)-t(n); t(n+a+1)-t(n+1), \ldots, t(n+2a-1)-t(n+a-1)\] are divisible by 4.
[i]Proposed by Gerhard Wöginger, Austria[/i]
2009 District Olympiad, 4
Let $ABC$ be an equilateral $ABC$. Points $M, N, P$ are located on the sides $AC, AB, BC$, respectively, such that $\angle CBM= \frac{1}{2} \angle AMN = \frac{1}{3} \angle BNP$ and $\angle CMP = 90 ^o$.
a) Show that $\vartriangle NMB$ is isosceles.
b) Determine $\angle CBM$.
2021 Taiwan APMO Preliminary First Round, 2
(a) Let the incenter of $\triangle ABC$ be $I$. We connect $I$ other $3$ vertices and divide $\triangle ABC$ into $3$ small triangles which has area $2,3$ and $4$. Find the area of the inscribed circle of $\triangle ABC$.
(b) Let $ABCD$ be a parallelogram. Point $E,F$ is on $AB,BC$ respectively. If $[AED]=7,[EBF]=3,[CDF]=6$, then find $[DEF].$ (Here $[XYZ]$ denotes the area of $XYZ$)
MathLinks Contest 5th, 4.2
Given is a unit cube in space. Find the maximal integer $n$ such that there are $n$ points, satisfying the following conditions:
(a) All points lie on the surface of the cube;
(b) No face contains all these points;
(c) The $n$ points are the vertices of a polygon.
2014 AMC 10, 22
In rectangle $ABCD$, $AB=20$ and $BC=10$. Let $E$ be a point on $\overline{CD}$ such that $\angle CBE=15^\circ$. What is $AE$?
$ \textbf{(A)}\ \dfrac{20\sqrt3}3\qquad\textbf{(B)}\ 10\sqrt3\qquad\textbf{(C)}\ 18\qquad\textbf{(D)}\ 11\sqrt3\qquad\textbf{(E)}\ 20 $
1991 India Regional Mathematical Olympiad, 3
A four-digit number has the following properties:
(a) It is a perfect square;
(b) Its first two digits are equal
(c) Its last two digits are equal.
Find all such four-digit numbers.
1975 Vietnam National Olympiad, 1
The roots of the equation $x^3 - x + 1 = 0$ are $a, b, c$. Find $a^8 + b^8 + c^8$.
2006 Iran MO (3rd Round), 5
Let $E$ be a family of subsets of $\{1,2,\ldots,n\}$ with the property that for each $A\subset \{1,2,\ldots,n\}$ there exist $B\in F$ such that $\frac{n-d}2\leq |A \bigtriangleup B| \leq \frac{n+d}2$. (where $A \bigtriangleup B = (A\setminus B) \cup (B\setminus A)$ is the symmetric difference). Denote by $f(n,d)$ the minimum cardinality of such a family.
a) Prove that if $n$ is even then $f(n,0)\leq n$.
b) Prove that if $n-d$ is even then $f(n,d)\leq \lceil \frac n{d+1}\rceil$.
c) Prove that if $n$ is even then $f(n,0) = n$
2011 AMC 10, 15
Roy bought a new battery-gasoline hybrid car. On a trip the car ran exclusively on its battery for the first 40 miles, then ran exclusively on gasoline for the rest of the trip, using gasoline at a rate of 0.02 gallons per mile. On the whole trip he averaged 55 miles per gallon. How long was the trip in miles?
$\textbf{(A)}\,140 \qquad\textbf{(B)}\,240 \qquad\textbf{(C)}\,440 \qquad\textbf{(D)}\,640 \qquad\textbf{(E)}\,840$
1990 Baltic Way, 5
Let $*$ be an operation, assigning a real number $a * b$ to each pair of real numbers $(a, b)$. Find an equation which is true (for all possible values of variables) provided the operation $*$ is commutative or associative and which can be false otherwise.