Olympiad problems

2023 CCA Math Bonanza, I10

Tags: CCA Math Bonanza

Bryan Ai has the following 8 numbers written from left to right on a sheet of paper: $$\textbf{1 4 1 2 0 7 0 8}$$ Now in each of the 7 gaps between adjacent numbers, Bryan Ai wants to place one of `$+$', `$-$', or `$\times$' inside that gap. Now, Bryan Ai wonders, if he picks a random placement out of the $3^7$ possible placements, what's the expected value of the expression (order of operations apply)? [i]Individual #10[/i]

2023 CCA Math Bonanza, TB4

Tags: CCA Math Bonanza , analytic geometry

Charlotte the cat is placed at the origin of the coordinate plane, such that the positive $x$ direction is pointing east, and the positive $y$ direction is pointing north. Then, every single lattice square (a unit square with vertices all on lattice points) in the first quadrant whose southwest vertex is a lattice point with both odd coordinates is completely removed. Charlotte can traverse the coordinate plane by drawing a segment between two valid vertices, given that they do not intersect a lattice square that has not been removed. Let $P_1$ and $P_2$ denote the distances of the first and second shortest paths Charlotte can take to $(5,7),$ respectively. Find $P_1-P_2.$ [i]Tiebreaker #4[/i]

2023 CCA Math Bonanza, L2.4

Tags: CCA Math Bonanza

A hundred people want to take a photo. They can stand in any number of rows from 1 to 100. Let $N$ be the number of possible photos they can take. What is the largest integer $k$ such that $2^k \mid N$? [i]Lightning 2.4[/i]

2023 CCA Math Bonanza, L3.1

Tags: CCA Math Bonanza

Joseph rolls a fair 6-sided dice repeatedly until he gets 3 of the same side in a row. What is the expected value of the number of times he rolls? [i]Lightning 3.1[/i]

2023 CCA Math Bonanza, I13

Tags: CCA Math Bonanza

Byan Rai has 1 red cup, 4 blue cups, 1 orange cup, 2 yellow cups, 3 green cups, 3 purple cups and 8 black cups in a box. Every second, Byan will pull out a random cup from the box and magically all other cups of the same color will disappear. What is the expected number of seconds it will take for Byan to pick a blue cup? [i]Individual #13[/i]

2016 CCA Math Bonanza, I15

Tags: CCA Math Bonanza , geometry

Let $ABC$ be a triangle with $AB=5$, $AC=12$ and incenter $I$. Let $P$ be the intersection of $AI$ and $BC$. Define $\omega_B$ and $\omega_C$ to be the circumcircles of $ABP$ and $ACP$, respectively, with centers $O_B$ and $O_C$. If the reflection of $BC$ over $AI$ intersects $\omega_B$ and $\omega_C$ at $X$ and $Y$, respectively, then $\frac{O_BO_C}{XY}=\frac{PI}{IA}$. Compute $BC$. [i]2016 CCA Math Bonanza Individual #15[/i]

2016 CCA Math Bonanza, T10

Tags: CCA Math Bonanza

Plusses and minuses are inserted in the expression \[\pm 1 \pm 2 \pm 3 \dots \pm 2016\] such that when evaluated the result is divisible by 2017. Let there be $N$ ways for this to occur. Compute the remainder when $N$ is divided by 503. [i]2016 CCA Math Bonanza Team #10[/i]

2023 CCA Math Bonanza, TB2

Tags: CCA Math Bonanza , geometry , 3D geometry , tetrahedron

How many ways are there to color a tetrahedron’s faces, edges, and vertices in red, green, and blue so that no face shares a color with any of its edges, and no edge shares a color with any of its endpoints? (Rotations and reflections are considered distinct.) [i]Tiebreaker #2[/i]

2023 CCA Math Bonanza, I7

Tags: CCA Math Bonanza

Of all positive integral solutions $(x,y,z)$ to the equation \[x^3+y^3+z^3-3xyz=607,\] compute the minimum possible value of $x+2y+3z.$ [i]Individual #7[/i]

2020 CCA Math Bonanza, I6

Tags: geometry , CCA Math Bonanza

Let $P$ be a point outside a circle $\Gamma$ centered at point $O$, and let $PA$ and $PB$ be tangent lines to circle $\Gamma$. Let segment $PO$ intersect circle $\Gamma$ at $C$. A tangent to circle $\Gamma$ through $C$ intersects $PA$ and $PB$ at points $E$ and $F$, respectively. Given that $EF=8$ and $\angle{APB}=60^\circ$, compute the area of $\triangle{AOC}$. [i]2020 CCA Math Bonanza Individual Round #6[/i]

2023 CCA Math Bonanza, I8

Tags: CCA Math Bonanza

Define a sequence of integers $a_1, a_2, \dots, a_k$ where every term $a_i \in \{1,2\}$, and let $S$ denote their sum. Another sequence of integers $t_1, t_2,\ldots, t_k$ is defined by \[t_i=\sqrt{a_i(S-a_i)},\] for all $t_i$. Suppose that $\sum_{1 \leq i \leq k} t_i=4000.$ Find the value of $\sum_{1 \leq i \leq k} a^2_i$. [i]Individual #8[/i]

2016 CCA Math Bonanza, L2.3

Tags: CCA Math Bonanza , inequalities

Let $ABC$ be a right triangle with $\angle{ACB}=90^{\circ}$. $D$ is a point on $AB$ such that $CD\perp AB$. If the area of triangle $ABC$ is $84$, what is the smallest possible value of $$AC^2+\left(3\cdot CD\right)^2+BC^2?$$ [i]2016 CCA Math Bonanza Lightning #2.3[/i]

2016 CCA Math Bonanza, T2

Tags: CCA Math Bonanza

Perry the painter wants to paint his floor, but he decides to leave a 1 foot border along the edges. After painting his floor, Perry notices that the area of the painted region is the same as the area of the unpainted region. Perry's floor measures $a$ x $b$ feet, where $a>b$ and both $a$ and $b$ are positive integers. Find all possible ordered pairs $(a, b)$. [i]2016 CCA Math Bonanza Team #2[/i]

2023 CCA Math Bonanza, I15

Tags: CCA Math Bonanza

Triangle $ABC$ has side lengths $AB=7, BC=8, CA=9.$ Define $M,N,P$ to be the midpoints of sides $BC,CA,AB,$ respectively. The circumcircles of $\triangle APN$ and $\triangle ABM$ intersect at another point $K.$ Find $NK.$ [i]Individual #15[/i]

2016 CCA Math Bonanza, L3.1

Tags: CCA Math Bonanza

How many 3-digit positive integers have the property that the sum of their digits is greater than the product of their digits? [i]2016 CCA Math Bonanza Lightning #3.1[/i]

2016 CCA Math Bonanza, I13

Tags: CCA Math Bonanza

Let $P(x)$ be a polynomial with integer coefficients, leading coefficient 1, and $P(0) = 3$. If the polynomial $P(x)^2 + 1$ can be factored as a product of two non-constant polynomials with integer coefficients, and the degree of $P$ is as small as possible, compute the largest possible value of $P(10)$. [i]2016 CCA Math Bonanza Individual #13[/i]

2023 CCA Math Bonanza, L5.2

Tags: CCA Math Bonanza

draNx rolls 1412 fair 6-sided dice. What is the probability the sum is in the range [4942, 5000]? Your score is determined by the function $max\{0, 20 - 200|A - E|\}$ where $A$ is the actual answer, and $E$ is your estimate. [i]Lightning 5.2[/i]

2016 CCA Math Bonanza, T9

Tags: CCA Math Bonanza , geometry , angle bisector

Let ABC be a triangle with $AB = 8$, $BC = 9$, and $CA = 10$. The line tangent to the circumcircle of $ABC$ at $A$ intersects the line $BC$ at $T$, and the circle centered at $T$ passing through $A$ intersects the line $AC$ for a second time at $S$. If the angle bisector of $\angle SBA$ intersects $SA$ at $P$, compute the length of segment $SP$. [i]2016 CCA Math Bonanza Team #9[/i]

2016 CCA Math Bonanza, L4.1

Tags: CCA Math Bonanza

Determine the remainder when $$2^6\cdot3^{10}\cdot5^{12}-75^4\left(26^2-1\right)^2+3^{10}-50^6+5^{12}$$ is divided by $1001$. [i]2016 CCA Math Bonanza Lightning #4.1[/i]

2016 CCA Math Bonanza, L1.1

Tags: CCA Math Bonanza

What is the sum of all the integers $n$ such that $\left|n-1\right|<\pi$? [i]2016 CCA Math Bonanza Lightning #1.1[/i]

2023 CCA Math Bonanza, L3.4

Tags: CCA Math Bonanza

Jonathan and Justin each flip a coin eight times. Jonathan and Justin get $m, n$ heads respectively. What is the probability that the difference of that $|m-n| \equiv 0 $ mod $4$? [i]Lightning 3.4[/i]

2023 CCA Math Bonanza, I12

Tags: CCA Math Bonanza

Find the sum of the real roots of $2x^4 + 4x^3 + 6x^2 + 4x - 4$. [i]Individual #12[/i]

2016 CCA Math Bonanza, I4

Tags: CCA Math Bonanza

The three digit number $n=CCA$ (in base $10$), where $C\neq A$, is divisible by $14$. How many possible values for $n$ are there? [i]2016 CCA Math Bonanza Individual #4[/i]

2016 CCA Math Bonanza, L4.2

Tags: CCA Math Bonanza , geometry , rectangle

Consider the $2\times3$ rectangle below. We fill in the small squares with the numbers $1,2,3,4,5,6$ (one per square). Define a [i]tasty[/i] filling to be one such that each row is [b]not[/b] in numerical order from left to right and each column is [b]not[/b] in numerical order from top to bottom. If the probability that a randomly selected filling is tasty is $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$, what is $m+n$? \begin{tabular}{|c|c|c|c|} \hline & & \\ \hline & & \\ \hline \end{tabular} [i]2016 CCA Math Bonanza Lightning #4.2[/i]

2016 CCA Math Bonanza, I8

Tags: CCA Math Bonanza

Let $f(x) = x^2 + x + 1$. Determine the ordered pair $(p,q)$ of primes satisfying $f(p) = f(q) + 242$. [i]2016 CCA Math Bonanza #8[/i]