Found problems: 66
2022 Middle European Mathematical Olympiad, 1
Find all functions $f: \mathbb R \to \mathbb R$ such that $$f(x+f(x+y))=x+f(f(x)+y)$$ holds for all real numbers $x$ and $y$.
2022 JHMT HS, 5
Suppose $\triangle JHU$ satisfies $JH = JU = 44$ and $HU = 32$. There is a unique circle passing through $U$ that is tangent to $\overline{JH}$ at its midpoint; let this circle intersect $\overline{JU}$ and $\overline{HU}$ again at points $X \neq U$ and $Y \neq U$, respectively. Let $Z$ be the unique point on $\overline{JH}$ such that $JZ = XU$. Compute the perimeter of quadrilateral $UXZY$.
2022 JHMT HS, 8
In equilateral $\triangle ABC$, point $D$ lies on $\overline{BC}$ such that the radius of the circumcircle $\Gamma_1$ of $\triangle ACD$ is $7$ and the radius of the incircle $\Gamma_2$ of $\triangle{ABD}$ is $2$. Suppose that $\Gamma_1$ and $\Gamma_2$ intersect at points $X$ and $Y$. Find $XY$.
2022 JHMT HS, 8
Let $P = (-4, 0)$ and $Q = (4, 0)$ be two points on the $x$-axis of the Cartesian coordinate plane, and let $X$ and $Y$ be points on the $x$-axis and $y$-axis, respectively, such that over all $Z$ on line $\overleftrightarrow{XY}$, the perimeter of $\triangle ZPQ$ has a minimum value of $25$. What is the smallest possible value of $XY^2$?
2022 JHMT HS, 5
Three congruent equilateral triangles $T_1$, $T_2$, and $T_3$ are stacked from left to right inside rectangle $JHMT$ such that the bottom left vertex of $T_1$ is $T$, the bottom side of $T_1$ lies on $\overline{MT}$, the bottom left vertex of $T_2$ is the midpoint of a side of $T_1$, the bottom left vertex of $T_3$ is the midpoint of a side of $T_2$, and the other two vertices of $T_3$ lie on $\overline{JH}$ and $\overline{HM}$, as shown below. Given that rectangle $JHMT$ has area $2022$, find the area of any one of the triangles $T_1$, $T_2$, or $T_3$.
[asy]
unitsize(0.111111111111111111cm);
real s = sqrt(4044/sqrt(75));
real l = 5s/2;
real w = s * sqrt(3);
pair J,H,M,T,V1,V2,V3,V4,V5,V6,V7,V8,C1,C2,C3;
J = (0,w);
H = (l,w);
M = (l,0);
T = (0,0);
V1 = (s,0);
V2 = (s/2,s * sqrt(3)/2);
V3 = (V1+V2)/2;
V4 = (3 * s/4+s,s * sqrt(3)/4);
V5 = (3 * s/4+s/2,s * sqrt(3)/4+s * sqrt(3)/2);
V6 = (V4+V5)/2;
V7 = (l,s * sqrt(3)/4+s * sqrt(3)/4);
V8 = (l-s/2,w);
C1 = (T+V1+V2)/3;
C2 = (V3+V4+V5)/3;
C3 = (V6+V7+V8)/3;
draw(J--H--M--T--cycle);
draw(V1--V2--T);
draw(V3--V4--V5--cycle);
draw(V6--V7--V8--cycle);
label("$J$", J, NW);
label("$H$", H, NE);
label("$M$", M, SE);
label("$T$", T, SW);
label("$T_1$", C1);
label("$T_2$", C2);
label("$T_3$", C3);
[/asy]
2022 JHMT HS, 6
For positive real numbers $a$ and $b,$ let $f(a,b)$ denote the real number $x$ such that area of the (non-degenerate) triangle with side lengths $a,b,$ and $x$ is maximized. Find
\[ \sum_{n=2}^{100}f\left(\sqrt{\tbinom{n}{2}},\sqrt{\tbinom{n+1}{2}}\right). \]
2022 JHMT HS, 1
If three of the roots of the quartic polynomial $f(x) = x^4 + ax^3 + bx^2 + cx + d$ are $0$, $2$, and $4$, and the sum of $a$, $b$, and $c$ is at most $12$, then find the largest possible value of $f(1)$.
2022 JHMT HS, 2
Four mutually externally tangent spherical apples of radius $4$ are placed on a horizontal flat table. Then, a spherical orange of radius $3$ is placed such that it rests on all the apples. Find the distance from the center of the orange to the table.
2022 JHMT HS, 2
Suppose that $f$ is a differentiable function such that $f(0) = 20$ and $|f'(x)| \leq 4$ for all real numbers $x$. Let $a$ and $b$ be real numbers such that [i]every[/i] such function $f$ satisfies $a \leq f(22) \leq b$. Find the smallest possible value of $|a| + |b|$.
2022 ISI Entrance Examination, 7
Let $$P(x)=1+2 x+7 x^{2}+13 x^{3}~,\qquad x \in \mathbb{R} .$$
Calculate for all $x \in \mathbb{R},$ $$\lim _{n \rightarrow \infty}\left(P\left(\frac{x}{n}\right)\right)^{n}$$
2022 JHMT HS, 1
Daredevil Darren challenges Forgetful Fred to spell "Johns Hopkins." Forgetful Fred will spell it correctly except for the 's's; there is a $\frac{1}{3}$ and $\frac{1}{4}$ chance that he will omit the 's' in the first and last names, respectively, with his mistakes being independent of each other. If Forgetful Fred spells the name correctly, then he is happy; otherwise, Daredevil Darren will present him with a dare, and there is a $\frac{9}{10}$ chance that Forgetful Fred will not be happy. Find the probability that Forgetful Fred will be happy.
2022 Kosovo Team Selection Test, 1
Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that for all real numbers $x$ and $y$,
$$f(x^2)+2f(xy)=xf(x+y)+yf(x).$$
[i]Proposed by Dorlir Ahmeti, Kosovo[/i]
2022 JHMT HS, 8
Let $\omega$ be a complex number satisfying $\omega^{2048} = 1$ and $\omega^{1024} \neq 1$. Find the unique ordered pair of nonnegative integers $(p, q)$ satisfying
\[ 2^p - 2^q = \sum_{0 \leq m < n \leq 2047} (\omega^m + \omega^n)^{2048}. \]
2022 JHMT HS, 8
Find the number of ways to completely cover a $2 \times 10$ rectangular grid of unit squares with $2 \times 1$ rectangles $R$ and $\sqrt{2}$ - $\sqrt{2}$ - $2$ triangles $T$ such that the following all hold:
[list]
[*] a placement of $R$ must have all of its sides parallel to the grid lines,
[*] a placement of $T$ must have its longest side parallel to a grid line,
[*] the tiles are non-overlapping, and
[*] no tile extends outside the boundary of the grid.
[/list]
(The figure below shows an example of such a tiling; consider tilings that differ by reflections to be distinct.)
[asy]
unitsize(1cm);
fill((0,0)--(10,0)--(10,2)--(0,2)--cycle, grey);
draw((0,0)--(10,0)--(10,2)--(0,2)--cycle);
draw((1,0)--(1,2));
draw((1,2)--(3,0));
draw((1,0)--(3,2));
draw((3,2)--(5,0));
draw((3,0)--(5,2));
draw((2,1)--(4,1));
draw((5,0)--(5,2));
draw((7,0)--(7,2));
draw((5,1)--(7,1));
draw((8,0)--(8,2));
draw((8,0)--(10,2));
draw((8,2)--(10,0));
[/asy]
2022 Indonesia Regional, 1
Let $A$ and $B$ be sets such that there are exactly $144$ sets which are subsets of either $A$ or $B$. Determine the number of elements $A \cup B$ has.
2022 JHMT HS, 6
Let $\lfloor x \rfloor$ denote the greatest integer less than or equal to $x$. Find the number of positive integers $m$ between $1$ and $2022$ inclusive such that
\[ \left\lfloor \frac{3^m}{11} \right\rfloor \]
is even.
2022 JHMT HS, 7
A spider sits on the circumference of a circle and wants to weave a web by making several passes through the circle's interior. On each pass, the spider starts at some location on the circumference, picks a destination uniformly at random from the circumference, and travels to that destination in a straight line, laying down a strand of silk along the line segment they traverse. After the spider does $2022$ of these passes (with each non-initial pass starting where the previous one ended), what is the expected number of points in the circle's interior where two or more non-parallel silk strands intersect?
2022 JHMT HS, 2
Erica intends to construct a subset $T$ of $S=\{ I,J,K,L,M,N \}$, but if she is unsure about including an element $x$ of $S$ in $T$, she will write $x$ in bold and include it in $T$. For example, $\{ I,J \},$ $\{ J,\mathbf{K},L \},$ and $\{ \mathbf{I},\mathbf{J},\mathbf{M},\mathbf{N} \}$ are valid examples of $T$, while $\{ I,J,\mathbf{J},K \}$ is not. Find the total number of such subsets $T$ that Erica can construct.
2022 JHMT HS, 3
Andy, Bella, and Chris are playing in a trivia contest. Andy has $21,200$ points, Bella has $23,600$ points, and Chris has $11,200$ points. They have reached the final round, which works as follows:
[list]
[*] they are given a hint as to what the only question of the round will be about;
[*] then, each of them must bet some amount of their points---this bet must be a nonnegative integer (a player does not know any of the other players' bets, and this bet cannot be changed later on);
[*] then, they will be shown the question, where they will have $30$ seconds to individually submit a response (a player does not know any of the other players' answers);
[*] finally, once time runs out, their responses and bets are revealed---if a player's response is correct, then the number of points they bet will be added to their score, otherwise, it will be subtracted from their score, and whoever ends up having the most points wins the contest (if there is a tie for the win, then the winner is randomly decided).
[/list]
Suppose that the contestants are currently deciding their bets based on the hint that the question will be about history. Bella knows that she will likely get the question wrong, but she also knows that Andy, who dislikes history, will definitely get it wrong. Knowing this, Bella wagers an amount that will guarantee her a win. Find the maximum number of points Bella could have ended up with.
2022 Indonesia Regional, 3
It is known that $x$ and $y$ are reals satisfying
\[ 5x^2 + 4xy + 11y^2 = 3. \]
Without using calculus (differentials/integrals), determine the maximum value of $xy - 2x + 5y$.
2022 JHMT HS, 7
Let $HOPKINS$ be an irregular convex heptagon (i.e., its angles and side lengths are all distinct, with the angles all having measure less than $180^{\circ}$) with area $1876$ such that all of its side lengths are greater than $5$, $OP=20$, and $KI=22$. Arcs with radius $2$ are drawn inside $HOPKINS$ with their centers at each of the vertices and their endpoints on the sides, creating circular sectors. Find the area of the region inside $HOPKINS$ but outside the sectors.
2022 Indonesia Regional, 2
(a) Determine a natural number $n$ such that $n(n+2022)+2$ is a perfect square.
[hide=Spoiler]In case you didn't realize, $n=1$ works lol[/hide]
(b) Determine all natural numbers $a$ such that for every natural number $n$, the number $n(n+a)+2$ is never a perfect square.
2022 JHMT HS, 10
Compute the exact value of
\[ \sum_{a=1}^{\infty}\sum_{b=1}^{\infty} \frac{\gcd(a, b)}{(a + b)^3}. \]
If necessary, you may express your answer in terms of the Riemann zeta function, $Z(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}$, for integers $s \geq 2$.
2022 JHMT HS, 9
Let $\{ a_n \}_{n=0}^{11}$ and $\{ b_n \}_{n=0}^{11}$ be sequences of real numbers. Suppose $a_0 = b_0 = -1$, $a_1 = b_1$, and for all integers $n \in \{2, 3, \ldots, 11\}$,
\begin{align*}
a_n & = a_{n-1} - (11 - n)^2(1 - (11 - (n - 1))^2)a_{n-2} \quad \text{and} \\
b_n & = b_{n-1} - (12 - n)^2(1 - (12 - (n - 1))^2)b_{n-2}.
\end{align*}
If $b_{11} = 2a_{11}$, then determine the value of $a_1$.
2022 Indonesia Regional, 5
Numbers $1$ to $22$ are written on a board. A "move" is a procedure of picking two numbers $a,b$ on the board such that $b \geq a+2$, then erasing $a$ and $b$ to be replaced with $a+1$ and $b-1$. Determine the maximum possible number of moves that can be done on the board.