id stringlengths 16 22 | problem stringlengths 57 1.26k | answer int64 0 991 | answer_str stringlengths 3 3 | tier stringclasses 4
values |
|---|---|---|---|---|
aimepp-aime-0001 | Let $X,Y,Z>1$ and $W>1$ satisfy $\log_X W=54$, $\log_Y W=24$, and $\log_{X^{3}Y^{2}Z^{2}}W=6$. Determine $\log_Z W$. | 72 | 072 | AIME |
aimepp-aime-0002 | For real $x$ with $-8\le x\le 25$, define $F(x)=|x+4|+2|x-10|+|x-12|+2|x-21|$. Find the minimum value of $F(x)$. | 38 | 038 | AIME |
aimepp-aime-0003 | Find the product of all real roots of \[x^2-5x+15=8\sqrt{x^2-5x}\]. | 225 | 225 | AIME |
aimepp-aime-0004 | A circle has radius $25$. Two parallel chords of lengths $48$ and $30$ lie on opposite sides of the center. Find the distance between the chords. | 27 | 027 | AIME |
aimepp-aime-0005 | Complex numbers $x$ and $y$ satisfy $x^2+y^2=91$ and $x^3+y^3=836$. Find the largest real value that $x+y$ can have. | 11 | 011 | AIME |
aimepp-aime-0006 | A sequence is defined by $a_1=8$, $a_2=27$, and $a_n=3a_{n-1}+6a_{n-2}$ for $n\ge3$. Find the remainder when $a_{829}$ is divided by $1000$. | 293 | 293 | AIME |
aimepp-aime-0007 | Of $31$ people seated around a circular table, three are chosen uniformly at random. The probability that at least two chosen people were adjacent is $\frac{p}{q}$ in lowest terms. Find $p+q$. | 173 | 173 | AIME |
aimepp-aime-0008 | Find the largest two-digit prime divisor of $\binom{164}{82}$. | 97 | 097 | AIME |
aimepp-aime-0009 | Find the minimum value of $\dfrac{36x^2\sin^2x+25}{x\sin x}$ for $0<x<\pi$. | 60 | 060 | AIME |
aimepp-aime-0010 | Consider $5$-digit decimal integers beginning with $8$ that have exactly one pair of equal digits, with every other digit occurring once. Find the remainder when the number of such integers is divided by $1000$. | 40 | 040 | AIME |
aimepp-aime-0011 | A rectangular box has side lengths $23$, $25$, and $h$. The centers of the three faces meeting at one vertex form a triangle of area $\frac{579}{8}$. Find $h$. | 2 | 002 | AIME |
aimepp-aime-0012 | In a circle, a diameter has the two-digit base-$10$ length $\overline{ab}_{10}$, and a perpendicular chord has length $\overline{ba}_{10}$. The diameter is longer, the two digits sum to $11$, and the distance from the chord to the center is rational. Find the diameter's value in base $10$. | 65 | 065 | AIME |
aimepp-aime-0013 | For every nonempty subset of $\{1,2,\ldots,14\}$, list its elements in decreasing order and alternately add and subtract, beginning with addition. Find the remainder when the sum of these alternating sums is divided by $1000$. | 688 | 688 | AIME |
aimepp-aime-0014 | In a circle, a diameter has the two-digit base-$9$ length $\overline{ab}_{9}$, and a perpendicular chord has length $\overline{ba}_{9}$. The diameter is longer, the two digits sum to $9$, and the distance from the chord to the center is rational. Find the diameter's value in base $10$. | 65 | 065 | AIME |
aimepp-aime-0015 | In a circle, a diameter has the two-digit base-$8$ length $\overline{ab}_{8}$, and a perpendicular chord has length $\overline{ba}_{8}$. The diameter is longer, the two digits sum to $7$, and the distance from the chord to the center is rational. Find the diameter's value in base $10$. | 35 | 035 | AIME |
aimepp-aime-0016 | The $8$ positive integer terms of an arithmetic sequence have total $420$, and the greatest term is $2$ times the least term. Find the greatest term. | 70 | 070 | AIME |
aimepp-aime-0017 | How many odd integers from $250$ through $999$ have no repeated decimal digit and have digit sum divisible by $5$? | 54 | 054 | AIME |
aimepp-aime-0018 | Find the area of a triangle with side lengths $10,13,13$. | 60 | 060 | AIME |
aimepp-aime-0019 | Positive integers $a\le b$ satisfy $\gcd(a,b)=3$ and $\operatorname{lcm}(a,b)=228$. Find the sum of all possible values of $a+b$. | 300 | 300 | AIME |
aimepp-aime-0020 | Let $X,Y,Z>1$ and $W>1$ satisfy $\log_X W=792$, $\log_Y W=352$, and $\log_{X^{3}YZ^{2}}W=96$. Determine $\log_Z W$. | 528 | 528 | AIME |
aimepp-aime-0021 | For real $x$ with $-21\le x\le 26$, define $F(x)=2|x+17|+3|x-8|+|x-18|+3|x-22|$. Find the minimum value of $F(x)$. | 102 | 102 | AIME |
aimepp-aime-0022 | A polynomial $P$ has degree at most $4$ and satisfies $P(0)=1$, $P(1)=-1$, $P(2)=-47$, $P(3)=-257$, $P(4)=-823$. Find the remainder when $P(5)$ is divided by $1000$. | 991 | 991 | AIME |
aimepp-aime-0023 | The nonreal roots of $z^2-7z+24=0$ are $\alpha$ and $\beta$. Find the remainder when $\alpha^{36}+\beta^{36}$ is divided by $1000$. | 977 | 977 | AIME |
aimepp-aime-0024 | A rectangular box has side lengths $3$, $8$, and $h$. The centers of the three faces meeting at one vertex form a triangle of area $\frac{61}{4}$. Find $h$. | 14 | 014 | AIME |
aimepp-aime-0025 | How many pairs of positive integers $(x,y)$ with $x\le y$ satisfy $\frac1x+\frac1y=\frac1{147}$? | 8 | 008 | AIME |
aimepp-aime-0026 | Three distinct vertices of a regular $10$-gon are chosen uniformly at random. The probability that they form an obtuse triangle is $\frac{m}{q}$ in lowest terms. Find $m+q$. | 3 | 003 | AIME |
aimepp-aime-0027 | How many subsets $S$ of $\{1,2,\ldots,10\}$ satisfy $\sum_{s\in S}s\equiv 4\pmod{7}$? The empty set is allowed. | 146 | 146 | AIME |
aimepp-aime-0028 | Angles $u$ and $v$ satisfy $\tan u+\tan v=34$ and $\cot u+\cot v=68$. Find $\tan(u+v)$. | 68 | 068 | AIME |
aimepp-aime-0029 | Find the smallest positive integer whose cube has last three digits $237$. | 933 | 933 | AIME |
aimepp-aime-0030 | A faculty has three departments: department 1 has 2 men and 3 women; department 2 has 3 men and 2 women; department 3 has 3 men and 3 women. A six-person committee must contain exactly two people from each department and exactly three men. How many committees are possible? | 558 | 558 | AIME |
aimepp-aime-0031 | A $6\times 5\times 9$ rectangular block is painted on all six faces and cut into unit cubes. How many unit cubes have exactly two painted faces? | 56 | 056 | AIME |
aimepp-aime-0032 | A faculty has three departments: department 1 has 2 men and 3 women; department 2 has 3 men and 2 women; department 3 has 4 men and 2 women. A six-person committee must contain exactly two people from each department and exactly three men. How many committees are possible? | 536 | 536 | AIME |
aimepp-aime-0033 | How many rectangles have all four vertices among the vertices of a regular $32$-gon? | 120 | 120 | AIME |
aimepp-aime-0034 | A polynomial $P$ has degree at most $4$ and satisfies $P(0)=-2$, $P(1)=8$, $P(2)=30$, $P(3)=40$, $P(4)=-10$. Find the remainder when $P(5)$ is divided by $1000$. | 808 | 808 | AIME |
aimepp-hard-0001 | Define $K_N=\displaystyle\prod_{j=1}^N(j!)^j$. Write $K_{236}=19^E U$, where $19\nmid U$. Let $Z$ be the least nonnegative residue of $E+U$ modulo $997$. Define $P(x)=x^{5} - 9 x^{4} + 349 x^{3} - 3 x^{2} + 7 x$ and $Q(x)=x^{4} + 692 x^{2} + 8 x - 63$. Find the least nonnegative residue modulo $1000$ of the resultant $... | 929 | 929 | AIME Hard |
aimepp-hard-0002 | Let $Z$ be the least nonnegative residue modulo $1000$ of the number of reduced fractions $a/b$ that satisfy $\frac{1}{5}<a/b<\frac{3}{5}$, $1\le b\le 554$, $\omega(b)=2$, and $a+b\equiv 1\pmod{7}$. Here $\omega(b)$ is the number of distinct prime divisors of $b$. Let $p=1361$, let $A$ and $B$ be the least nonnegative ... | 835 | 835 | AIME Hard |
aimepp-hard-0003 | A partition of $\{1,2,\ldots,9\}$ has a crossing quadruple when $a<b<c<d$, the numbers $a,c$ lie in one block, and $b,d$ lie in a different block. Let $Z$ be the least nonnegative residue modulo $1000$ of the number of set partitions that have exactly $4$ blocks, exactly $0$ singleton blocks, and a number of crossing q... | 284 | 284 | AIME Hard |
aimepp-hard-0004 | Let $C_{253}=\frac1{254}\binom{506}{253}$ be a Catalan number. Write $C_{253}=5^e u$ with $5\nmid u$. Let $Z$ be the least nonnegative residue of $u+84e$ modulo $1000$. Write $Z=100d_2+10d_1+d_0$ with $0\le d_0,d_1,d_2\le9$. Let $\lambda=(12+d_2+d_1+d_0,8+d_1+d_0,4+d_0,4,2)$. Find the least nonnegative residue modulo $... | 192 | 192 | AIME Hard |
aimepp-hard-0005 | Consider standard Young tableaux of skew shape $(4,2,2,2,2)/(0,0,0,0,0)$. A descent is an integer $i$ for which the entry $i+1$ lies in a lower row than the entry $i$. Let $Z$ be the least nonnegative residue modulo $1000$ of the number of such tableaux that have exactly $5$ descents. Put $L=18+(Z\bmod 13)$ and, for $1... | 24 | 024 | AIME Hard |
aimepp-hard-0006 | A tournament is formed by orienting every edge of the complete graph on $7$ labeled vertices. Let $Z$ be the least nonnegative residue modulo $1000$ of the number of tournaments that have sorted outdegree sequence $(1, 1, 1, 4, 4, 5, 5)$ and exactly $3$ directed $3$-cycles. A Markov chain has transition matrix $\frac1{... | 771 | 771 | AIME Hard |
aimepp-hard-0007 | Let $\theta_k=\frac{2\pi k}{19}$ for $1\le k\le 18$. If $\displaystyle\sum_{k=1}^{18}\left(\frac1{(8-2\cos\theta_k)^2}+\frac{2}{8-2\cos\theta_k}\right)=\frac pq$ in lowest terms, let $Z$ be the least nonnegative residue of $p+q$ modulo $1000$. Let $m=8+(Z\bmod 9)$. Form a wheel with hub $0$ and rim vertices $1,2,\ldots... | 525 | 525 | AIME Hard |
aimepp-hard-0008 | Let $Z$ be the least nonnegative residue modulo $1000$ of the number of $5\times5$ zero-one matrices that have every row sum equal to $2$ or $3$, exactly $1$ rows of sum $3$, no zero column, rank $4$ over $\mathbb F_2$, trace $3$, an anti-diagonal sum congruent to $0$ modulo $3$, and exactly $5$ columns of odd sum. Fin... | 702 | 702 | AIME Hard |
aimepp-hard-0009 | Define $K_N=\displaystyle\prod_{j=1}^N(j!)^j$. Write $K_{195}=11^E U$, where $11\nmid U$. Let $Z$ be the least nonnegative residue of $E+U$ modulo $997$. Put $L=Z+30$. Define $G(n,0)=1$, $G(0,k)=0$ for $k>0$, and $G(n,k)=G(n-1,k)+7^{n-k}G(n-1,k-1)$. Find the least nonnegative residue modulo $1000$ of $G(L,11)+3G(L,10)$... | 208 | 208 | AIME Hard |
aimepp-hard-0010 | On an $8\times 8$ board, row $i$ contains exactly the first $b_i$ allowed squares, where $(b_1,\ldots,b_8)=(1, 2, 3, 7, 8, 8, 8, 8)$. Let $Z$ be the least nonnegative residue modulo $1000$ of the number of ways to place $8$ nonattacking rooks, one in each row and column, so that exactly $4$ rooks lie on the main diagon... | 447 | 447 | AIME Hard |
aimepp-hard-0011 | Let $Y(x)$ be the unique formal power series with constant term $0$ satisfying $Y=x(1+Y)^{2}(1+Y^2)^{3}(1-Y)^{-1}$. Let $Z$ be the least nonnegative residue modulo $997$ of the coefficient of $x^{36}$ in $Y(x)^{10}$. Let $R$ be the least nonnegative residue modulo $10^{10}-1$ of $19^{Z^2+3Z+86}+31^{Z+86}+4510209742$, a... | 135 | 135 | AIME Hard |
aimepp-hard-0012 | Let $N=6452641$. For all coprime positive integer pairs $(x,y)$ with $x>y$ and $x^2-xy+y^2=N$, consider those satisfying $2x-y\equiv 11\pmod{13}$. Let $Z$ be the least nonnegative residue modulo $1000$ of the sum of $3x+y$ over all such pairs. Define $u_0=81$ and $u_1=458$, and for $k\ge 0$ define $u_{k+2}$ by $u_{k+2}... | 545 | 545 | AIME Hard |
aimepp-hard-0013 | Let $Y(x)$ be the unique formal power series with constant term $0$ satisfying $Y=x(1+Y)^{2}(1+Y^2)^{3}(1-Y)^{-1}$. Let $Z$ be the least nonnegative residue modulo $991$ of the coefficient of $x^{43}$ in $Y(x)^{10}$. Let $p=2129$, let $A$ and $B$ be the least nonnegative residues modulo $p$ of $611Z+422$ and $1387Z+870... | 723 | 723 | AIME Hard |
aimepp-hard-0014 | Over the ring $(\mathbb Z/1000\mathbb Z)[x]$, let $c_0+c_1x+\cdots+c_6x^6$ be the remainder when $(x^2+x^4+3)^{154655378506}$ is divided by the monic polynomial $P(x)=x^{7} - 5 x^{6} - 6 x^{5} - 5 x^{4} - 2 x^{3} - 5 x - 5$. Let $Z$ be the least nonnegative residue of $9c_0+1c_1+2c_2+7c_3+5c_4+2c_5+9c_6$ modulo $1000$.... | 611 | 611 | AIME Hard |
aimepp-hard-0015 | Let $Z$ be the least nonnegative residue modulo $1000$ of the number of reduced fractions $a/b$ that satisfy $\frac{1}{5}<a/b<\frac{3}{5}$, $1\le b\le 466$, $\omega(b)=1$, and $a+b\equiv 6\pmod{13}$. Here $\omega(b)$ is the number of distinct prime divisors of $b$. Let $n=7+(Z\bmod 4)$ and let $M=(m_{ij})_{0\le i,j<n}$... | 171 | 171 | AIME Hard |
aimepp-hard-0016 | Let $Z$ be the least nonnegative residue modulo $1000$ of the number of $5\times5$ zero-one matrices that have every row sum equal to $2$ or $3$, exactly $5$ rows of sum $3$, no zero column, rank $4$ over $\mathbb F_2$, trace $2$, an anti-diagonal sum congruent to $0$ modulo $3$, and exactly $5$ columns of odd sum. Put... | 69 | 069 | AIME Hard |
aimepp-hard-0017 | An urn contains $11$ red, $11$ blue, and $10$ green balls. Balls are drawn uniformly without replacement until the number drawn of some color first reaches its threshold: $5$ for red, $4$ for blue, and $2$ for green. The probability that the stopping color is green and the total number of draws is congruent to $3$ modu... | 205 | 205 | AIME Hard |
aimepp-hard-0018 | Triangle $ABC$ has side lengths $BC=21$, $CA=15$, and $AB=16$. An interior point $P$ has barycentric coordinates $(8:7:3)$, and $Q$ is the isotomic conjugate of $P$. The perpendiculars from $Q$ to $BC,CA,AB$ meet those lines at $D,E,F$, respectively. If $([DEF]/[ABC])^2=p/q$ in lowest terms, let $Z$ be the least nonneg... | 376 | 376 | AIME Hard |
aimepp-hard-0019 | For a positive integer $t$, let $J_2(t)=t^2\prod_{p\mid t}(1-p^{-2})$. Let $N=780$. Let $Z$ be the least nonnegative residue modulo $997$ of $\displaystyle\sum (d+1)J_2(N/d)$, where the sum is over squarefree divisors $d$ of $N$ such that $d\equiv 1\pmod{11}$ and $\Omega(d)\equiv 1\pmod2$. A Markov chain has transition... | 115 | 115 | AIME Hard |
aimepp-hard-0020 | The sequence $(a_n)$ is defined by $(a_0,a_1,\ldots,a_5)=(17, 16, -12, -18, 13, 10)$ and $a_{n+6}=8a_{n+5}-2a_{n+4}+7a_{n+3}-6a_{n+2}-5a_{n+1}+8a_n$ for $n\ge0$. Let $M$ be the $5\times5$ matrix whose $(i,j)$ entry is $a_{8798202908543+i+j}-5a_{8798202908543+i+j+1}$, where $0\le i,j\le4$. Let $Z$ be the least nonnegati... | 475 | 475 | AIME Hard |
aimepp-hard-0021 | The sequence $(a_n)$ is defined by $(a_0,a_1,\ldots,a_5)=(-18, -13, 14, 11, -13, -9)$ and $a_{n+6}=5a_{n+5}+7a_{n+4}+8a_{n+3}-8a_{n+2}+7a_{n+1}+3a_n$ for $n\ge0$. Let $M$ be the $5\times5$ matrix whose $(i,j)$ entry is $a_{2431472563692+i+j}-1a_{2431472563692+i+j+1}$, where $0\le i,j\le4$. Let $Z$ be the least nonnegat... | 967 | 967 | AIME Hard |
aimepp-hard-0022 | Let $N=38909$. Among all pairs of positive integers $(x,y)$ satisfying $x<y$, $x^2+y^2=N$, and $\gcd(x,y)=1$, retain only those for which $x+2y\equiv 1\pmod{13}$. Let $Z$ be the least nonnegative residue modulo $1000$ of $\displaystyle\sum(2x+3y)$ over the retained pairs. Put $L=Z+50$. Let $S(n,k)$ be a Stirling number... | 65 | 065 | AIME Hard |
aimepp-hard-0023 | For $0\le i,j\le5$, define $m_{ij}=\binom{29+i+j}{23+2i-j}+3\binom{29+i+j}{23-i+2j}+3\binom{29+i+j}{23+i-j-2}$, where a binomial coefficient is $0$ when its lower index is outside its usual range. If $M=(m_{ij})_{0\le i,j\le5}$, let $Z$ be the least nonnegative residue of $\det M$ modulo $991$. A Markov chain has trans... | 149 | 149 | AIME Hard |
aimepp-hard-0024 | On an $8\times 8$ board, row $i$ contains exactly the first $b_i$ allowed squares, where $(b_1,\ldots,b_8)=(2, 3, 6, 8, 8, 8, 8, 8)$. Let $Z$ be the least nonnegative residue modulo $1000$ of the number of ways to place $8$ nonattacking rooks, one in each row and column, so that exactly $0$ rooks lie on the main diagon... | 145 | 145 | AIME Hard |
aimepp-hard-0025 | Let $C_{267}=\frac1{268}\binom{534}{267}$ be a Catalan number. Write $C_{267}=5^e u$ with $5\nmid u$. Let $Z$ be the least nonnegative residue of $u+28e$ modulo $1000$. Put $L=Z+70$. Let $S(n,k)$ be a Stirling number of the second kind, and let $A(n,j)$ be the number of permutations of $\{1,\ldots,n\}$ with exactly $j$... | 900 | 900 | AIME Hard |
aimepp-hard-0026 | For $1\le n\le 72793$, set $a_n=\lfloor n\sqrt{12}\rfloor$ and $b_n=\lfloor n/\sqrt{12}\rfloor$. Let $Z$ be the least nonnegative residue modulo $1000$ of the sum of all $n$ such that $a_n\equiv 2\pmod{11}$, $b_n\equiv 4\pmod{6}$, and $n\equiv 1\pmod2$. Let $L=Z+52$ and $P=\displaystyle\prod_{j=1}^L(8j+7)$. Write $P=5^... | 546 | 546 | AIME Hard |
aimepp-hard-0027 | Let $p=3079$. A number $g$ with $1<g<p$ is called admissible when it is a primitive root modulo $p$, the sum of its base-$11$ digits is congruent to $5$ modulo $11$, and the sum of the base-$11$ digits of its least positive inverse modulo $p$ is congruent to $9$ modulo $10$. Let $Z$ be the least nonnegative residue mod... | 460 | 460 | AIME Hard |
aimepp-hard-0028 | Two walkers move simultaneously on the lattice, each taking one unit north or east step per second. The lower walker goes from $(0,0)$ to $(9,11)$, and the upper walker goes from $(0,2)$ to $(9,13)$. After every equal number of steps, their current vertices must be distinct. A turn is a change of direction by either wa... | 55 | 055 | AIME Hard |
aimepp-hard-0029 | Let $N=30240$. Consider nondecreasing $4$-tuples of integers $2\le a_1\le\cdots\le a_4$ whose product is $N$. Among those having exactly $2$ even entries, greatest common divisor $1$, and $a_1+\cdots+a_4\equiv 4\pmod{6}$, let $Z$ be the least nonnegative residue modulo $1000$ of the sum of all possible values of $a_4$.... | 17 | 017 | AIME Hard |
aimepp-hard-0030 | A sphere $\Sigma$ is orthogonal to four spheres having, respectively, center $(-8,5,-7)$ and squared radius $\frac{221}{2}$; center $(-5,-1,4)$ and squared radius $\frac{133}{2}$; center $(-6,8,-6)$ and squared radius $\frac{177}{2}$; and center $(1,3,-5)$ and squared radius $\frac{11}{2}$. If the center of $\Sigma$ is... | 413 | 413 | AIME Hard |
aimepp-hard-0031 | In triangle $ABC$, the side lengths opposite $A,B,C$ are $26,32,12$, respectively. An interior point $P$ has barycentric coordinates $(5:2:2)$, and $Q$ is the isogonal conjugate of $P$. If $PQ^2=p/q$ in lowest terms, let $Z$ be the least nonnegative residue of $p+q$ modulo $1000$. In the quotient ring $(\mathbb Z/1000\... | 9 | 009 | AIME Hard |
aimepp-hard-0032 | Let $r_1,r_2,r_3,r_4$ be the roots of $P(x)=x^{4} + 7 x^{3} - 3 x^{2} - 3 x + 7$. For $i<j$ define $u_{ij}=r_ir_j+4$. Let $Z$ be the least nonnegative residue modulo $1000$ of $\displaystyle\sum_{i<j}\left(u_{ij}^{1282}-7u_{ij}^{1281}\right)$. Let $L=Z+47$ and $P=\displaystyle\prod_{j=1}^L(16j+9)$. Write $P=5^eU$ with ... | 945 | 945 | AIME Hard |
aimepp-hard-0033 | Write the base-$7$ expansion of $1/83$ as $0.\overline{d_1d_2\cdots d_h}_{7}$ with the shortest possible period, and read subscripts cyclically modulo $h$. For each $j$, let $B_j$ be the base-$7$ integer with digits $d_j,d_{j+1},\ldots,d_{j+6}$. Let $Z$ be the least nonnegative residue modulo $1000$ of the sum of all $... | 828 | 828 | AIME Hard |
aimepp-hard-0034 | Consider ordered pairs $(x,y)$ of least positive residues modulo $1296$ satisfying $xy\equiv 701\pmod{1296}$, $x+y\equiv 8\pmod{14}$, and $x+2y\equiv 4\pmod{5}$. Let $Z$ be the least nonnegative residue modulo $1000$ of $\displaystyle\sum(x+3y)$ over all such pairs. Find the least nonnegative residue modulo $1000$ of t... | 124 | 124 | AIME Hard |
aimepp-hard-0035 | Let $N=329509$. For all coprime positive integer pairs $(x,y)$ with $x>y$ and $x^2-xy+y^2=N$, consider those satisfying $2x-y\equiv 7\pmod{9}$. Let $Z$ be the least nonnegative residue modulo $1000$ of the sum of $3x+y$ over all such pairs. A Markov chain has transition matrix $\frac1{10}\begin{pmatrix}4 & 2 & 2 & 2\\2... | 887 | 887 | AIME Hard |
aimepp-hard-0036 | Let $C_{315}=\frac1{316}\binom{630}{315}$ be a Catalan number. Write $C_{315}=3^e u$ with $3\nmid u$. Let $Z$ be the least nonnegative residue of $u+81e$ modulo $1000$. Define $u_0=402$ and $u_1=422$, and for $k\ge 0$ define $u_{k+2}$ by $u_{k+2}\equiv 19u_{k+1}-17u_k+7(-1)^k\pmod{1000}$, with every $u_k$ chosen from $... | 192 | 192 | AIME Hard |
aimepp-hard-0037 | Let $p=2017$. A number $g$ with $1<g<p$ is called admissible when it is a primitive root modulo $p$, the sum of its base-$10$ digits is congruent to $7$ modulo $9$, and the sum of the base-$10$ digits of its least positive inverse modulo $p$ is congruent to $3$ modulo $9$. Let $Z$ be the least nonnegative residue modul... | 759 | 759 | AIME Hard |
aimepp-hard-0038 | A sphere $\Sigma$ is orthogonal to four spheres having, respectively, center $(7,5,4)$ and squared radius $\frac{521}{4}$; center $(6,3,7)$ and squared radius $\frac{617}{4}$; center $(-6,-3,-6)$ and squared radius $\frac{133}{4}$; and center $(-6,4,7)$ and squared radius $\frac{441}{4}$. If the center of $\Sigma$ is $... | 800 | 800 | AIME Hard |
aimepp-hard-0039 | Over the ring $(\mathbb Z/1000\mathbb Z)[x]$, let $c_0+c_1x+\cdots+c_6x^6$ be the remainder when $(x^2+x^3-2)^{110460296634}$ is divided by the monic polynomial $P(x)=x^{7} + x^{6} + 4 x^{5} - 4 x^{4} + 4 x^{3} - 5 x^{2} + 2 x + 2$. Let $Z$ be the least nonnegative residue of $1c_0+7c_1+4c_2+4c_3+9c_4+2c_5+7c_6$ modulo... | 864 | 864 | AIME Hard |
aimepp-hard-0040 | Over the ring $(\mathbb Z/1000\mathbb Z)[x]$, let $c_0+c_1x+\cdots+c_6x^6$ be the remainder when $(x^1+x^5+2)^{267703890501}$ is divided by the monic polynomial $P(x)=x^{7} + 5 x^{6} + 6 x^{5} - 6 x^{4} + 2 x^{3} - 5 x^{2} + x + 4$. Let $Z$ be the least nonnegative residue of $9c_0+2c_1+2c_2+4c_3+7c_4+1c_5+4c_6$ modulo... | 720 | 720 | AIME Hard |
aimepp-hard-0041 | Let $\theta_k=\frac{2\pi k}{9}$ for $1\le k\le 8$. If $\displaystyle\sum_{k=1}^{8}\left(\frac1{(8-2\cos\theta_k)^2}+\frac{4}{8-2\cos\theta_k}\right)=\frac pq$ in lowest terms, let $Z$ be the least nonnegative residue of $p+q$ modulo $1000$. Let $p=1889$, set $x_0=Z$, and for $k\ge0$ let $x_{k+1}$ be the least nonnegati... | 923 | 923 | AIME Hard |
aimepp-hard-0042 | A plane partition in an $3\times 5\times 4$ box is a $3\times 5$ array $(p_{ij})$ of integers from $0$ through $4$ that is weakly decreasing across every row and down every column. Let $Z$ be the least nonnegative residue modulo $1000$ of the number of such arrays that have total sum congruent to $2$ modulo $5$, diagon... | 36 | 036 | AIME Hard |
aimepp-hard-0043 | A simple graph on $6$ labeled vertices is chosen uniformly from all connected such graphs. The probability that it has exactly $2$ vertices of odd degree and exactly $4$ triangles is $p/q$ in lowest terms. Let $Z$ be the least nonnegative residue of $p+q$ modulo $1000$. Find the least nonnegative residue modulo $1000$ ... | 13 | 013 | AIME Hard |
aimepp-hard-0044 | The eight vertices of a cube are colored red, green, or blue, using the colors exactly $2, 5, 1$ times, respectively. Two colorings are considered the same when a rotation of the cube carries one to the other. Let $Z$ be the least nonnegative residue modulo $1000$ of the number of equivalence classes that have exactly ... | 276 | 276 | AIME Hard |
aimepp-hard-0045 | Let $Z$ be the least nonnegative residue modulo $1000$ of the number of permutations $\pi$ of $\{1,2,\ldots,9\}$ that have exactly $3$ cycles in their disjoint-cycle decomposition, exactly $2$ excedances, exactly $1$ fixed points, and major index congruent to $6$ modulo $7$. Put $L=18+(Z\bmod 13)$ and, for $1\le j\le L... | 743 | 743 | AIME Hard |
aimepp-hard-0046 | Triangle $ABC$ has side lengths $BC=17$, $CA=12$, and $AB=26$. An interior point $P$ has barycentric coordinates $(6:8:7)$, and $Q$ is the isotomic conjugate of $P$. The perpendiculars from $Q$ to $BC,CA,AB$ meet those lines at $D,E,F$, respectively. If $([DEF]/[ABC])^2=p/q$ in lowest terms, let $Z$ be the least nonneg... | 687 | 687 | AIME Hard |
aimepp-hard-0047 | A partition of $\{1,2,\ldots,9\}$ has a crossing quadruple when $a<b<c<d$, the numbers $a,c$ lie in one block, and $b,d$ lie in a different block. Let $Z$ be the least nonnegative residue modulo $1000$ of the number of set partitions that have exactly $6$ blocks, exactly $5$ singleton blocks, and a number of crossing q... | 508 | 508 | AIME Hard |
aimepp-hard-0048 | In triangle $ABC$, the side lengths opposite $A,B,C$ are $33,33,13$, respectively. An interior point $P$ has barycentric coordinates $(5:4:3)$, and $Q$ is the isogonal conjugate of $P$. If $PQ^2=p/q$ in lowest terms, let $Z$ be the least nonnegative residue of $p+q$ modulo $1000$. Let $A=\begin{pmatrix}3 & 4 & 1 & 4\\2... | 295 | 295 | AIME Hard |
aimepp-hard-0049 | For a positive integer $t$, let $J_2(t)=t^2\prod_{p\mid t}(1-p^{-2})$. Let $N=2012472$. Let $Z$ be the least nonnegative residue modulo $997$ of $\displaystyle\sum (d+1)J_2(N/d)$, where the sum is over squarefree divisors $d$ of $N$ such that $d\equiv 6\pmod{13}$ and $\Omega(d)\equiv 0\pmod2$. Let $A=\begin{pmatrix}0 &... | 334 | 334 | AIME Hard |
aimepp-hard-0050 | Let $a_n$ be defined by $\displaystyle\sum_{n\ge0}a_nx^n=\frac{(1+x^8)^{5}(1+x^7+x^{14})^{5}}{(1-x^2)^{3}(1-x^5)^{2}}$. Let $Z$ be the least nonnegative residue modulo $983$ of $\displaystyle\sum_{\substack{0\le n\le 116\n\equiv 6\pmod{9}}}a_n$. Let $n=7+(Z\bmod 4)$ and let $M=(m_{ij})_{0\le i,j<n}$, where $m_{ij}$ is ... | 720 | 720 | AIME Hard |
aimepp-hard-0051 | The graph $C_5\square P_15$ has $15$ cyclic rows of $5$ vertices; adjacent vertices in each row are joined, including the wraparound pair, and corresponding vertices in consecutive rows are joined. Let $Z$ be the least nonnegative residue modulo $1000$ of the number of independent sets that contain exactly $5$ vertices... | 36 | 036 | AIME Hard |
aimepp-hard-0052 | A tetrahedron $ABCD$ has squared edge lengths $AB^2=36$, $AC^2=128$, $AD^2=54$, $BC^2=68$, $BD^2=102$, and $CD^2=230$. Let $V$ be its volume and $R$ its circumradius. Write $R^2=p/q$ in lowest terms. Let $Z$ be the least nonnegative residue modulo $1000$ of $288V^2+p+q$. Let $R$ be the least nonnegative residue modulo ... | 30 | 030 | AIME Hard |
aimepp-hard-0053 | Let $p=5843$. A number $g$ with $1<g<p$ is called admissible when it is a primitive root modulo $p$, the sum of its base-$5$ digits is congruent to $4$ modulo $10$, and the sum of the base-$5$ digits of its least positive inverse modulo $p$ is congruent to $5$ modulo $7$. Let $Z$ be the least nonnegative residue modulo... | 320 | 320 | AIME Hard |
aimepp-hard-0054 | A tetrahedron $ABCD$ has squared edge lengths $AB^2=49$, $AC^2=208$, $AD^2=116$, $BC^2=145$, $BD^2=109$, and $CD^2=68$. Let $V$ be its volume and $R$ its circumradius. Write $R^2=p/q$ in lowest terms. Let $Z$ be the least nonnegative residue modulo $1000$ of $288V^2+p+q$. Define $u_0=616$ and $u_1=471$, and for $k\ge 0... | 536 | 536 | AIME Hard |
aimepp-hard-0055 | Let $r_1,r_2,r_3,r_4$ be the complex roots, counted with multiplicity, of $P(x)=x^{4} + x^{3} + x^{2} - 3 x + 3$. For the six numbers $s_{ij}=r_i+r_j$ with $1\le i<j\le4$, set $T=\displaystyle\sum_{i<j}\left(\frac{1}{(11-s_{ij})^2}+4\frac{1}{11-s_{ij}}\right)$. If $T=\frac pq$ in lowest terms with $q>0$, let $Z$ be the... | 234 | 234 | AIME Hard |
aimepp-hard-0056 | For $0\le i,j\le5$, define $m_{ij}=\binom{37+i+j}{20+2i-j}+4\binom{37+i+j}{20-i+2j}+2\binom{37+i+j}{20+i-j-2}$, where a binomial coefficient is $0$ when its lower index is outside its usual range. If $M=(m_{ij})_{0\le i,j\le5}$, let $Z$ be the least nonnegative residue of $\det M$ modulo $991$. Write $Z=100d_2+10d_1+d_... | 0 | 000 | AIME Hard |
aimepp-hard-0057 | Let $p=1579$. A number $g$ with $1<g<p$ is called admissible when it is a primitive root modulo $p$, the sum of its base-$11$ digits is congruent to $6$ modulo $11$, and the sum of the base-$11$ digits of its least positive inverse modulo $p$ is congruent to $3$ modulo $6$. Let $Z$ be the least nonnegative residue modu... | 0 | 000 | AIME Hard |
aimepp-hard-0058 | The sequence $(a_n)$ is defined by $(a_0,a_1,\ldots,a_5)=(-3, 14, -17, -17, 3, 13)$ and $a_{n+6}=8a_{n+5}-5a_{n+4}+2a_{n+2}+7a_{n+1}+a_n$ for $n\ge0$. Let $M$ be the $5\times5$ matrix whose $(i,j)$ entry is $a_{3800044329232+i+j}-3a_{3800044329232+i+j+1}$, where $0\le i,j\le4$. Let $Z$ be the least nonnegative residue ... | 573 | 573 | AIME Hard |
aimepp-hard-0059 | Let $Z$ be the least nonnegative residue modulo $1000$ of the number of length-$42$ words $a_1a_2\cdots a_42$ over $\{0,1,2,3\}$ that contain none of the blocks $00$, $123$, or $232$, contain exactly $38$ occurrences of $3$, satisfy $\sum_{i=1}^42 i a_i\equiv 3\pmod7$, and have a number of indices with $a_{i+1}>a_i$ co... | 133 | 133 | AIME Hard |
aimepp-hard-0060 | Consider ordered pairs $(x,y)$ of least positive residues modulo $12960$ satisfying $xy\equiv 5729\pmod{12960}$, $x+y\equiv 6\pmod{8}$, and $x+2y\equiv 2\pmod{9}$. Let $Z$ be the least nonnegative residue modulo $1000$ of $\displaystyle\sum(x+3y)$ over all such pairs. Put $L=18+(Z\bmod 13)$ and, for $1\le j\le L$, put ... | 534 | 534 | AIME Hard |
aimepp-hard-0061 | The sequence $(a_n)$ is defined by $(a_0,a_1,\ldots,a_5)=(16, -18, 18, -17, -16, 16)$ and $a_{n+6}=-8a_{n+5}+4a_{n+2}-7a_{n+1}-a_n$ for $n\ge0$. Let $M$ be the $5\times5$ matrix whose $(i,j)$ entry is $a_{917841055891+i+j}-4a_{917841055891+i+j+1}$, where $0\le i,j\le4$. Let $Z$ be the least nonnegative residue of $\det... | 910 | 910 | AIME Hard |
aimepp-hard-0062 | Let $N=1440$. Consider nondecreasing $4$-tuples of integers $2\le a_1\le\cdots\le a_4$ whose product is $N$. Among those having exactly $2$ even entries, greatest common divisor $1$, and $a_1+\cdots+a_4\equiv 7\pmod{9}$, let $Z$ be the least nonnegative residue modulo $1000$ of the sum of all possible values of $a_4$. ... | 228 | 228 | AIME Hard |
aimepp-hard-0063 | Let $C_{243}=\frac1{244}\binom{486}{243}$ be a Catalan number. Write $C_{243}=11^e u$ with $11\nmid u$. Let $Z$ be the least nonnegative residue of $u+67e$ modulo $1000$. Let $M=3080$ and let $R$ be the least nonnegative residue modulo $M$ of $(10Z+239)^2+1(10Z+239)$. Let $\mathcal S$ be the set of residues $z\in\{0,1,... | 236 | 236 | AIME Hard |
aimepp-hard-0064 | Let $p=4597$. A number $g$ with $1<g<p$ is called admissible when it is a primitive root modulo $p$, the sum of its base-$8$ digits is congruent to $1$ modulo $5$, and the sum of the base-$8$ digits of its least positive inverse modulo $p$ is congruent to $0$ modulo $10$. Let $Z$ be the least nonnegative residue modulo... | 30 | 030 | AIME Hard |
aimepp-hard-0065 | In triangle $ABC$ with $BC=36$, $CA=36$, and $AB=31$, points $D,E,F$ lie on $BC,CA,AB$, respectively, and satisfy $BD:DC=2:4$, $CE:EA=1:5$, and $AF:FB=3:3$. The circles $(ADE)$, $(BEF)$, and $(CFD)$ have radical center $X$. If the power of $X$ with respect to the circumcircle of $ABC$ is $p_0/q_0$ in lowest terms with ... | 590 | 590 | AIME Hard |
aimepp-hard-0066 | For a positive integer $t$, let $J_2(t)=t^2\prod_{p\mid t}(1-p^{-2})$. Let $N=26583700$. Let $Z$ be the least nonnegative residue modulo $997$ of $\displaystyle\sum (d+1)J_2(N/d)$, where the sum is over squarefree divisors $d$ of $N$ such that $d\equiv 0\pmod{10}$ and $\Omega(d)\equiv 1\pmod2$. Write $Z=100d_2+10d_1+d_... | 320 | 320 | AIME Hard |
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