Found problems: 37
2024 JHMT HS, 9
Let $N \in \{10, 11, \ldots, 99\}$ be a two-digit positive integer. Compute the number of values of $N$ for which the last two digits in the decimal expansion of $N^{21}$ are the digits of $N$ in the same order.
2024 JHMT HS, 12
Let $\{ a_n \}_{n=0}^{\infty}$, $\{ b_n \}_{n=0}^{\infty}$, and $\{ c_n \}_{n=0}^{\infty}$ be sequences of real numbers such that for all $k\geq 1$,
\begin{align*}
a_k&=\left\lfloor \sqrt{2}+\frac{k-1}{2024} \right\rfloor+a_{k-1} \\
b_k+c_k&=1 \\
a_{k-1}b_k&=a_kc_k.
\end{align*}
Suppose that $a_0=1$, $b_0=2$, and $c_0=3$. Given that $\sqrt2\approx1.4142$, compute
\[ \sum_{k=1}^{2024}(a_kb_k-a_{k-1}c_k). \]
2024 Iranian Geometry Olympiad, 4
Point $P$ is inside the acute triangle $\bigtriangleup ABC$ such that $\angle BPC=90^{\circ}$ and $\angle BAP=\angle PAC$. Let $D$ be the projection of $P$ onto the side $BC$. Let $M$ and $N$ be the incenters of the triangles $\bigtriangleup ABD$ and $\bigtriangleup ADC$ respectively. Prove that the quadrilateral $BMNC$ is cyclic.
[i]Proposed by Hussein Khayou - Syria[/i]
2024 JHMT HS, 2
Let $Q$ be a quadratic polynomial with a unique zero. Suppose $Q(12)=Q(16)$ and $Q(20)=24$. Compute $Q(28)$.
2024 JHMT HS, 10
One triangular face $F$ of a tetrahedron $\mathcal{T}$ has side lengths $\sqrt{5}$, $\sqrt{65}$, and $2\sqrt{17}$. The other three faces of $\mathcal{T}$ are right triangles whose hypotenuses coincide with the sides of $F$. There exists a sphere inside $\mathcal{T}$ tangent to all four of its faces. Compute the radius of this sphere.
2024 JHMT HS, 1
Compute the smallest positive integer $N$ for which $N \cdot 2^{2024}$ is a multiple of $2024$.
2024 JHMT HS, 7
Let $N_6$ be the answer to problem 6.
Given positive integers $n$ and $a$, the $n$[i]th tetration of[/i] $a$ is defined as
\[ ^{n}a=\underbrace{a^{a^{\cdot^{\cdot^{\cdot^a}}}}}_{n \text{ times}}. \]
For example, $^{4}2=2^{2^{2^2}}=2^{2^4}=2^{16}=65536$. Compute the units digit of $^{2024}N_6$.
2024 Pan-American Girls’ Mathematical Olympiad, 5
Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that
$f(f(x+y) - f(x)) + f(x)f(y) = f(x^2) - f(x+y),$
for all real numbers $x, y$.
2024 JHMT HS, 6
Let $N_5$ be the answer to problem 5.
Triangle $JHU$ satisfies $JH=N_5$ and $JU=6$. Point $X$ lies on $\overline{HU}$ such that $\overline{JX}$ is an altitude of $\triangle{JHU}$, point $Y$ is the midpoint of $\overline{JU}$, and $\overline{JX}$ and $\overline{HY}$ intersect at $Z$. Assume that $\triangle{HZX}$ is similar to $\triangle{JZY}$ (in this vertex order). Compute the area of $\triangle{JHU}$.
2024 JHMT HS, 8
Let $N_7$ be the answer to problem 7.
Each side of a regular $N_7$-gon is colored with a single color from a set of two given colors. Two colorings that can be obtained from one another by a rotation or a reflection of the entire figure are considered the same. Compute the number of possible different colorings.
2024 JHMT HS, 10
Let $N_9$ be the answer to problem 9.
In a rainforest, there is a row of nine rocks labeled from $1$ to $N_9$. A gecko is standing on rock $1$. The gecko jumps according to following rules:
[list]
[*] if it is on rock $1$, then it will jump with equal probability to any of the other rocks.
[*] if it is on rock $R$ and $R$ is prime, then it will jump to rock $N_9$.
[*] if it is on rock $4$, then it will jump with equal probability to rock $1$ or rock $6$.
[*] if it is on rock $R$ and $R$ is composite with $4<R<N_9$, then it will jump with equal probability to rock $Q$ or $S$, where $Q$ is the greatest composite number less than $R$, and $S$ is the smallest composite number greater than $R$.
[*] if it is on rock $N_9$, then it stops jumping.
[/list]
The expected number of jumps the gecko will take to reach rock $N_9$ is $\tfrac{p}{q}$, where $p$ are $q$ relatively prime positive integers. Compute $p+q$.
2024 JHMT HS, 4
Let $N_3$ be the answer to problem 3.
Compute the sum of all real solutions $x$ to the equation
\[ 50^x+72^x+(N_3)^x=800^x. \]
2024 JHMT HS, 13
In prism $JHOPKINS$, quadrilaterals $JHOP$ and $KINS$ are parallel and congruent bases that are kites, where $JH = JP = KI = KS$ and $OH = OP = NI = NS$; the longer two sides of each kite have length $\tfrac{4 + \sqrt{5}}{2}$, and the shorter two sides of each kite have length $\tfrac{5 + \sqrt{5}}{4}$. Assume that $\overline{JK}$, $\overline{HI}$, $\overline{ON}$, and $\overline{PS}$ are congruent edges of $JHOPKINS$ perpendicular to the planes containing $JHOP$ and $KINS$. Vertex $J$ is part of a regular pentagon $JAZZ'Y$ that can be inscribed in prism $JHOPKINS$ such that $A \in \overline{HI}$, $Z \in \overline{NI}$, $Z' \in \overline{NS}$, $Y \in \overline{PS}$, $AI = YS$, and $ZI = Z'S$. Compute the height of $JHOPKINS$ (that is, the distance between the bases).
2024 JHMT HS, 15
Let $\ell = 1$, $M = 23$, $N = 45$, and $u = 67$. Compute the number of ordered pairs of nonnegative integers $(X, Y)$ with $X \leq M - \ell$ and $Y \leq N + u$ such that the sum
\[ \sum_{k=\ell}^{u} \binom{X + k}{M}\cdot\binom{Y - k}{N} \]
is divisible by $89$ (for integers $a$ and $b$, define the binomial coefficient $\tbinom{a}{b}$ to be the number of $b$-element subsets of any given $a$-element set, which is $0$ when $a < 0$, $b < 0$, or $b > a$).
2024 JHMT HS, 16
Let $N_{15}$ be the answer to problem 15.
For a positive integer $x$ expressed in base ten, let $x'$ be the result of swapping its first and last digits (for example, if $x = 2024$, then $x' = 4022$). Let $C$ be the number of $N_{15}$-digit positive integers $x$ with a nonzero leading digit that satisfy the property that both $x$ and $x'$ are divisible by $11$ (note: $x'$ is allowed to have a leading digit of zero). Compute the sum of the digits of $C$ when $C$ is expressed in base ten.
2024 JHMT HS, 8
Points $A$, $B$, $C$, and $D$ lie on a circle $\Gamma$, in that order, with $AB=5$ and $AD=3$. The angle bisector of $\angle ABC$ intersects $\Gamma$ at point $E$ on the opposite side of $\overleftrightarrow{CD}$ as $A$ and $B$. Assume that $\overline{BE}$ is a diameter of $\Gamma$ and $AC=AE$. Compute $DE$.
2024 JHMT HS, 7
Compute the sum of all real solutions $\alpha$ (in radians) to the equation
\[ |\sin\alpha|=\left\lfloor \frac{\alpha}{20} \right\rfloor. \]
2024 JHMT HS, 11
Call a positive integer [i]convenient[/i] if its digits can be partitioned into two collections of contiguous digits whose element sums are $7$ and $11$. For example, $3456$ is convenient, but $4247$ is not. Compute the number of convenient positive integers less than or equal to $10^5$.
2024 AIME, 7
Let $N$ be the greatest four-digit integer with the property that whenever one of its digits is changed to $1$, the resulting number is divisible by $7$. Let $Q$ and $R$ be the quotient and remainder, respectively, when $N$ is divided by $1000$. Find $Q+R$.
2024 JHMT HS, 5
Compute the positive difference between the two solutions to the equation $2x^2-28x+9=0$.
2024 JHMT HS, 6
Compute the number of nonempty subsets $S$ of $\{ 1,2,3,4,5,6,7,8,9,10 \}$ such that the median of $S$ is an element of $S$.
2024 JHMT HS, 15
Let $N_{14}$ be the answer to problem 14.
Rectangle $ABCD$ has area $\sqrt{2N_{14}}$. Points $E$, $F$, $G$, and $H$ lie on the rays $\overrightarrow{AB}$, $\overrightarrow{BC}$, $\overrightarrow{CD}$, and $\overrightarrow{DA}$, respectively, such that $EFGH$ is a rectangle with area $2\sqrt{2N_{14}}$ that contains all of $ABCD$ in its interior. If
\[ \tan\angle AEH = \tan\angle BFE = \tan\angle CGF = \tan\angle DHG = \sqrt{\frac{1}{48}}, \]
then $EG=\tfrac{m\sqrt{n}}{p}$, where $m$, $n$, and $p$ are positive integers, $m$ and $p$ are relatively prime, and $n$ is not divisible by the square of any prime. Compute $m + n + p$.
2024 JHMT HS, 5
Triangle $JHU$ has side lengths $JH=13$, $HU=14$, and $JU=15$. Point $X$ lies on $\overline{HU}$ such that $\triangle{JHX}$ and $\triangle{JUX}$ have equal perimeters. Compute $JX^2$.
2024 JHMT HS, 4
Let $x$ be a real number satisfying
\[ \sqrt[3]{125-x^3}-\sqrt[3]{27-x^3}=7. \]
Compute $|\sqrt[3]{125-x^3}+\sqrt[3]{27-x^3}|$.
2024 JHMT HS, 12
Let $N_{11}$ be the answer to problem 11.
Concave heptagon $HOPKINS$, where $180^\circ<\angle HOP<270^\circ$, has area $N_{11}$, and $HP=NI\sqrt{24}$. Suppose that $HONS$ and $OPKI$ are congruent squares. Compute the common area of each of these squares.